MATH10200705001 Neeldhara Misra - Homi Bhabha National ...
MATH10200705001 Neeldhara Misra - Homi Bhabha National ...
MATH10200705001 Neeldhara Misra - Homi Bhabha National ...
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KERNELS FOR THE F-DELETION PROBLEM<br />
By<br />
<strong>Neeldhara</strong> <strong>Misra</strong><br />
The Institute of Mathematical Sciences, Chennai.<br />
Athesissubmittedtothe<br />
Board of Studies in Mathematical Sciences<br />
In partial fulfillment of the requirements<br />
For the Degree of<br />
DOCTOR OF PHILOSOPHY<br />
of<br />
HOMI BHABHA NATIONAL INSTITUTE<br />
September 2011
<strong>Homi</strong> <strong>Bhabha</strong> <strong>National</strong> Institute<br />
Recommendations of the Viva Voce Board<br />
As members of the Viva Voce Board, we recommend that the dissertation prepared<br />
by <strong>Neeldhara</strong> <strong>Misra</strong> entitled “Kernels for the F-Deletion Problem” may<br />
be accepted as fulfilling the dissertation requirement for the Degree of Doctor of<br />
Philosophy.<br />
Chairman : Chair of committe<br />
Convener : Conv of Committe<br />
Member : Member 1 of committe<br />
Member : Member 2 of committe<br />
Date :<br />
Date :<br />
Date :<br />
Date :<br />
Final approval and acceptance of this dissertation is contingent upon the candidate’s<br />
submission of the final copies of the dissertation to HBNI.<br />
IherebycertifythatIhavereadthisdissertationpreparedundermydirection<br />
and recommend that it may be accepted as fulfilling the dissertation requirement.<br />
Guide : Venkatesh Raman<br />
Date :
DECLARATION<br />
I, hereby declare that the investigation presented in the thesis<br />
has been carried out by me. The work is original and the<br />
work has not been submitted earlier as a whole or in part for<br />
adegree/diplomaatthisoranyotherInstitutionorUniversity.<br />
<strong>Neeldhara</strong> <strong>Misra</strong>
ACKNOWLEDGEMENTS<br />
Ihavebeenextremelyfortunateinhavingenjoyedtheguidanceandcareofmy<br />
advisor, Prof. Venkatesh Raman. His presence has been a source of inspiration and<br />
strength like no other, and every opportunity of interaction has been a treasured<br />
learning experience. More importantly than leading me up to successes, he found<br />
me the motivation that I needed to last me through the inevitably larger number<br />
of failures. I will never cease be amazed at his expertise and teaching, and my own<br />
stroke of luck for having had the chance to witness it from close quarters.<br />
I am also deeply obliged to Prof. Saket Saurabh for his relentless efforts in<br />
advancing my comprehension of a number of things, technical and non-technical<br />
alike. He has always been incredibly accessible and it is thanks to his infinite<br />
patience that I eventually got around to messing with things that I would otherwise<br />
keep at a safe distance. I also learned from him to value the pursuit as much as, and<br />
irrespective of, the associated discovery. The energy and intensity that he brings<br />
to research is nothing short of magical, and the experience has been a privilege.<br />
Prof. Mike Fellows and Frances Rosamond have my eternal gratitude for taking<br />
care of me at various stages, and ensuring that I have nothing less than the time<br />
of my life! They have been very generous in their enthusiastic support, and I have<br />
certainly received from them much more than I deserve.<br />
I would like to thank Prof. Fedor Fomin, whom I had an opportunity to<br />
visit at the University of Bergen — the stay was intensely enriching. He was<br />
very generous in with sharing his profound experience, and the discussions were<br />
delightful. Special thanks are due to Prof. Pinar Heggernes, Pim van’t Hof, Daniel<br />
Marx, and Yngve Villanger for a particularly memorable collaboration. I learned<br />
agooddealfromDanielLokshtanov,andIamthankfulforhistimeandvaluable<br />
insights.<br />
Thanks to Somnath Sikdar, Geevarghese Philip, and M. S. Ramanujan, the<br />
typical day was always something to look forward to!<br />
I would like to thank Abhimanyu M. Ambalath, Radheshyam Balasundaram,<br />
Chintan Rao H., Venkata Koppula, Matthias Mnich, N. S. Narayanaswamy, and<br />
Bal Sri Shankar, for many exciting discussions. I also enjoyed very much a number<br />
of discussions with Prof. Ronojoy Adhikari - thank you!
I will be forever grateful to all the faculty at the Department of Theoretical<br />
Computer Science. Any understanding that I might have developed about Computer<br />
Science certainly is thanks to their patient, creative, and inspiring teaching. I<br />
am equally thankful to all my friends and colleagues for being tirelessly supportive.<br />
The work leading to this thesis executed at IMSc and the University of Bergen. I<br />
would like to thank both institutions for the financial support, excellent infrastructure<br />
and wonderful research environments. Special thanks to the administrative<br />
staff, who have always been especially helpful.<br />
My parents happen to be my life, universe, and everything. For chasing my<br />
dreams with me, in true no-matter-what style, thanks very much indeed. I will<br />
add that more can be said that goes without saying, and is therefore left unsaid.<br />
5
Abstract<br />
In this thesis, we use the parameterized framework for the design and analysis<br />
of algorithms for NP-complete problems. This amounts to studying the parameterized<br />
version of the classical decision version. Herein, the classical language<br />
appended with a secondary measure called a “parameter”. The central notion in<br />
parameterized complexity is that of fixed-parameter tractability, which means given<br />
an instance (x, k) of a parameterized language L, deciding whether (x, k) ∈ L in<br />
time f(k) · p(|x|), where f is an arbitrary function of k alone and p is a polynomial<br />
function. The notion of kernelization formalizes preprocessing or data reduction,<br />
and refers to polynomial time algorithms that transform any given input into an<br />
equivalent instance whose size is bounded as a function of the parameter alone.<br />
The center of our attention in this thesis is the F-Deletion problem, a vastly<br />
general question that encompasses many fundamental optimization problems as<br />
special cases. In particular, we provide evidence supporting a conjecture about the<br />
kernelization complexity of the problem, and this work branches off in a number of<br />
directions, leading to results of independent interest. We also study the Colorful<br />
Motifs problem, a well-known question that arises frequently in practice. Our<br />
investigation demonstrates the hardness of the problem even when restricted to<br />
very simple graph classes.<br />
The F-Deletion Problem Let F be a finite family of graphs. The F-Deletion<br />
problem takes as input a graph G on n vertices, and a positive integer k. The<br />
question is whether it is possible to delete at most k vertices from G such that<br />
the remaining graph contains no graph from F as a minor. This question encompasses<br />
fundamental problems such as Vertex Cover (consider F consisting of<br />
the graph with two vertices and one edge) or Feedback Vertex Set (the set<br />
F consists of a cycle with three vertices). A number of other deletion-based optimization<br />
problems also turn out to be special cases of Planar F-Deletion, for<br />
instance: Diamond Hitting Set, Outerplanar Deletion Set, Pathwidth<br />
r-Deletion Set and Treewidth r-Deletion Set.<br />
The general F-Deletion problem is NP-complete. From the parameterized<br />
perspective, by one of the most well-known consequences of the celebrated Graph<br />
Minor theory of Robertson and Seymour, the F-Deletion problem is fixed-<br />
6
parameter tractable for every finite set of forbidden minors. It is conjectured that<br />
the F-Deletion problem admits a polynomial kernel if, and only if, F contains<br />
aplanargraph. WerefertotheF-Deletion question when restricted to the case<br />
when F contains a planar graph as the Planar F-Deletion problem.<br />
We obtain a number of kernelization results for the Planar F-Deletion problem.<br />
We give a linear vertex kernel on graphs excluding t-claw K 1,t , the star with<br />
t leves, as an induced subgraph, where t is a fixed integer, and a obtain polynomial<br />
kernels for the case when F contains graph θ c as a minor for a fixed integer<br />
c. The graph θ c consists of two vertices connected by c parallel edges. Even<br />
though this may appear to be a very restricted class of problems it already encompasses<br />
well-studied problems such as Vertex Cover, Feedback Vertex Set<br />
and Diamond Hitting Set. The generic kernelization algorithm is based on a<br />
non-trivial application of protrusion techniques, previously used only for problems<br />
on topological graph classes. We also demonstrate an approximation algorithm<br />
achieving an approximation ratio of O(log 3/2 OPT), where OPT is the size of an<br />
optimal solution on general undirected graphs. The approximation algorithm is<br />
used crucially in the kernelization routines.<br />
We also show an algorithm for Planar F-Deletion that runs in time 2 O(k log k) n 2 ,<br />
improving substantially from what was known previously, 2 2O(k log k) n O(1) . Since<br />
Planar F-Deletion is a generalization of Vertex Cover, wehavethatitcannot<br />
be solved in time 2 o(k) n O(1) , unless Exponential Time Hypothesis fails, so our<br />
algorithm is also quite close to being optimal. This algorithm uses the technique<br />
of iterative compression, and the instance encountered at every iteration is subject<br />
to kernelization. The intermediate kernelization algorithm is quite non-trivial and<br />
requires reformulating some of the fundamental machinery that was used in all the<br />
other situations. Also, towards the kernel, we show a “decomposition lemma” that<br />
asserts that any graph of constant treewidth can essentially be partitioned into a<br />
number of protrusions. We believe this result to be of independent interest.<br />
We also consider the complementary “packing” question: we wish to maximize<br />
the number of vertex (or edge) disjoint minor models of graphs in F. We show<br />
that the packing number is closely related to the size of the optimal F-hitting<br />
set in the special case when F = {θ c }. Independently, but on a related note, we<br />
show two lower bounds, demonstrating that the problem of finding if there are at<br />
least k vertex-disjoint minor models of θ c is unlikely to admit a polynomial kernel<br />
7
parameterized by k, andthisisalsotruefortheproblemofcheckingifthereare<br />
at least k vertex-disjoint cycles of odd length (again parameterized by k).<br />
Colorful Motifs We study the problem of Colorful Motifs on various graph<br />
classes. We prove that the problem of Colorful Motifs restricted to superstars<br />
is NP-complete. Further, we show NP-completeness on graphs of diameter two.<br />
We apply this result towards settling the classical complexity of Connected<br />
Dominating Set on graphs of diameter two — specifically, we show that it is<br />
NP-complete. Further, we show that on graphs of diameter two, the problem is<br />
NP-complete and is unlikely to admit a polynomial kernel.<br />
Next, we show that obtaining polynomial kernels for Colorful Motifs on<br />
comb graphs is infeasible, but we show the existence of n polynomial kernels.<br />
Further, we study the problem of Colorful Motifs on trees, where we observe<br />
that the natural strategies for many polynomial kernels are not successful. For<br />
instance, we show that “guessing” a root vertex, which helped in the case of comb<br />
graphs, fails as a strategy because the Rooted Colorful Motif problem has<br />
no polynomial kernels on trees.<br />
8
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1 Introduction<br />
1 Introduction<br />
1 Introduction<br />
i T 2 Let F be a finite set of graphs. In an p-F-Deletio<br />
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this is the Vertex ∗ Department of Informatics, University of Bergen, N-5020 Bergen, Norway. fomin<br />
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† {neeldhara|gphilip|saket}@imsc.res.in.<br />
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CA 92093-0404, USA, dlokshtanov@cs.ucsd.edu<br />
‡ We use prefix p to distinguish the parameterized version of the problem.<br />
The Institute of Mathematical 1 <br />
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{neeldhara|gphilip|saket}@imsc.res.in.<br />
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O ∗ (2 |M| )
G =(V, E) k ∈ N c : V → [k]<br />
k<br />
G T k c T <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
k <br />
<br />
<br />
F + <br />
<br />
+ <br />
θ c <br />
F
+ <br />
+
+ <br />
+
G H F H J <br />
k l <br />
<br />
V(G) G E(G) <br />
M G G <br />
G <br />
u, v, x v G <br />
v d(v) ∆(G) <br />
G δ(G) G<br />
n ∈ N [n] {1, . . . , n}<br />
<br />
<br />
<br />
<br />
G (V, E) V E ⊆ V × V V<br />
E
G ′ G V(G ′ ) ⊆ V(G) E(G ′ ) ⊆ E(G) G ′<br />
G E(G ′ )={{u, v} ∈ E(G) | u, v ∈ V(G ′ )} <br />
S ⊆ V(G) S G[S] <br />
V(G)\S G\S N(S) <br />
S V(G) \ S S<br />
(u, v) G <br />
u v <br />
G H G <br />
θ c <br />
<br />
<br />
H G <br />
φ : V(H) → 2 V(G) v ∈ V(H) G[φ(v)] <br />
v, u ∈ V(H) φ(u) ∩ φ(v) =∅ (u, v) ∈ E(H) <br />
(u ′ ,v ′ ) ∈ E(G) u ′ ∈ φ(u) v ′ ∈ φ(v)<br />
G, H G ′ G <br />
H G G ′ H G ′ <br />
H G G ′ H G<br />
<br />
c ∈ N M G <br />
θ c G M T 1 T 2 S c <br />
T 1 T 2 <br />
C C C<br />
C H H H/∈ C<br />
k G <br />
k G <br />
G <br />
γ(G) G <br />
k <br />
k 1 <br />
n n
V(G) <br />
<br />
B G <br />
<br />
<br />
(l × l) <br />
<br />
<br />
⋆ <br />
S <br />
<br />
<br />
⋆ B B <br />
<br />
<br />
⋆ G G<br />
<br />
G<br />
<br />
(l × l) <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
T 1 ,T 2 ,...,T n T <br />
V(T i ) ∩ V(T j ) ≠ ∅ ∀i, j.
a, b, c T<br />
<br />
V(T 1 ) ∩ V(T 2 ) ∩ ...∩ V(T n ) ≠ ∅.<br />
a, b, c T <br />
⋂ s∈S V(T s) ≠ ∅.<br />
S = {i : T i a, b, c}.<br />
P 1 (a, b) T P 2 (b, c) T P 3 <br />
(a, c) T T V(P 1 ) ∩ V(P 2 ) ∩ V(P 3 )<br />
x <br />
<br />
T s s ∈ S P 1 ,P 2 P 3 x ∈ V(T s )<br />
s ∈ S <br />
n<br />
n = 1, 2 <br />
n T n T 1 ,T 2 ,...,T n<br />
T <br />
a, b, c <br />
a ∈ V(T 1 ) ∩ V(T 2 ) ∩ ···∩ V(Tn1),<br />
b ∈ V(T 2 ) ∩ V(T 3 ) ∩ ···∩ V(T n )<br />
c ∈ V(T 1 ) ∩ V(T n ).
T i a, b, c <br />
<br />
V(T 1 ) ∩ V(T 2 ) ∩ ...∩ V(T n ) ≠ ∅.<br />
<br />
<br />
L Σ ∗ × N Σ <br />
(x, k) k <br />
<br />
(x, k) f(k) · p(|x|) f <br />
k p <br />
<br />
<br />
Q ⊆ Σ ∗ × N (x, k) ∈ Σ ∗ × N <br />
|x| + k (x ′ ,k ′ ) ∈ Σ ∗ × N (x, k) ∈ Q <br />
(x ′ ,k ′ ) ∈ Q |x ′ | + k ′ g(k) g <br />
g g <br />
Q <br />
<br />
<br />
<br />
<br />
<br />
<br />
G G (T, X = {X t } t∈V(T ) ) <br />
<br />
∪ t∈V(T ) X t = V(G)<br />
{x, y} ∈ E(G) t ∈ V(T) {x, y} ⊆ X t <br />
v ∈ V(G) T {t | v ∈ X t }
( max t∈V(T ) |X t | ) − 1 G <br />
G (T, X)<br />
T r X r = ∅<br />
T <br />
<br />
t t ′ X t ⊃ X t ′ |X t | =<br />
|X t ′| + 1<br />
t t ′ X t ⊂ X t ′ |X t | =<br />
|X t ′| − 1<br />
t t 1 t 2 X t = X t1 = X t2 <br />
t t X t = ∅<br />
r T <br />
G <br />
O(|V(G)| + |E(G)|) <br />
G t ∪ t ′X ′ t <br />
t ′ t t H t G t [V(G t ) \ X t ]<br />
v ∈ G T {t | v ∈ X t } T v <br />
<br />
<br />
B ⊆ V(G) G T B := ∪{T v | v ∈<br />
V(B)} T<br />
T B <br />
T <br />
T T 1<br />
T 2 i = 1, 2 B i ∪{v ∈ B | v ∈ X(T i )} X(T i ) <br />
T i B <br />
G (u, v) u ∈ B 1 v ∈ B 2 <br />
X uv u v <br />
T 1 T 2 T 1<br />
T 2
G χ(G) (G) +1 χ(G) <br />
G<br />
|V(G)| |V(G)| = n n 2 <br />
χ(G) (G)+1 n <br />
G |V(G)| = n+1 |V(G)| 2 <br />
v d(v) (G) T G <br />
T ′ V(T) V(T ′ ) t <br />
t ′ t <br />
v ∈ X t \ X t ′ X t t <br />
X t X t ′ <br />
t <br />
T v X t <br />
v <br />
v d(v) (G)<br />
v G d(v) (G). <br />
χ(G \{v}) (G \{v})+1,<br />
(G) +1 d(v) (G) <br />
X ((G) +1) G \{v} v <br />
X X <br />
((G) +1) G \{v} v X <br />
χ(G) (G)+1<br />
G S ⊆ V(G) ∂ G (S) S <br />
V(G) \ S S ⊆ V(G) S N G (S) =<br />
∂ G (V(G) \ S) <br />
+ <br />
r G X ⊆ V(G) <br />
r G (G[X]) r |∂(X)| r
∨ ∧ ¬ ⇔ ⇒<br />
∀ ∃ <br />
<br />
u ∈ U u U <br />
d ∈ D d D <br />
inc(d, u) d u <br />
d u<br />
adj(u, v) u v u u<br />
v <br />
<br />
<br />
<br />
p <br />
p Π G k <br />
S k <br />
P Π (G, S) <br />
<br />
<br />
<br />
φ <br />
t <br />
G t <br />
S G φ f(t, |φ|)|V(G)|
∨ ∧ ¬ ⇔ ⇒<br />
∀ ∃ <br />
<br />
u ∈ U u U <br />
d ∈ D d D <br />
inc(d, u) d u <br />
d u<br />
adj(u, v) u v u u<br />
v <br />
<br />
<br />
<br />
p <br />
p Π G k <br />
S k <br />
P Π (G, S) <br />
<br />
<br />
<br />
φ <br />
t <br />
G t <br />
S G φ f(t, |φ|)|V(G)|
I NP <br />
I ′ |I ′ | < |I| <br />
NP <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
k <br />
<br />
k <br />
p(k) k <br />
p(k) p(k) =O(k) <br />
<br />
<br />
2k 67k<br />
O(k 2 ) <br />
<br />
<br />
<br />
<br />
k <br />
k <br />
d <br />
4k <br />
2 k
φ <br />
k <br />
φ 3 n m <br />
<br />
<br />
k m/2 <br />
k <br />
<br />
m 2k n 6k <br />
<br />
<br />
d <br />
d <br />
<br />
dd C d <br />
U k <br />
U ′ ⊆ U k U ′ <br />
C<br />
<br />
<br />
k Y S 1 ,S 2 ···S k S i ∩ S j = Y <br />
i ≠ j S i \ Y <br />
<br />
F <br />
U s |F| >s!(k − 1) s F k <br />
F U
d<br />
d O(k d d!d 2 ) (U, C,k)<br />
d (U, C ′ ,k ′ ) |C ′ | O(k d d!d)<br />
<br />
C S = {S 1 , ··· ,S k+1 }<br />
k + 1 C k <br />
Y S k<br />
C ′ = C \ (S ∪ Y) (U, C,k) (U, C ′ ,k) <br />
<br />
d ′ ∈ {1, ··· ,d} <br />
k + 1 <br />
d ′ ∈ {1, ··· ,d} O(k d′ d ′ !) d <br />
O(k d d!d)<br />
<br />
<br />
<br />
<br />
<br />
<br />
G =(V, E) k <br />
V ′ ⊆ V k (u, v) ∈ E<br />
u ∈ V ′ v ∈ V ′ <br />
2 U = V C = {{u, v}|(uv) ∈<br />
E} 4k 2 8k 2 <br />
4k <br />
<br />
G =(V, E) V<br />
C H R C H <br />
<br />
C
C R N[C] ∩ R = ∅<br />
E ′ C H E ′ <br />
|H| G ′ =(C ∪ H, E ′ ) <br />
H<br />
<br />
<br />
G =(V, E) I ⊆ V |N(I)| <<br />
|I| (C, H, R) G C ⊆ I <br />
O(m + n) G I<br />
<br />
<br />
(C, H, R) G =(V, E) G<br />
k G ′ = G[R] k ′ = k−|H|<br />
G V ′ k G <br />
|H| C H H |V ′ ∩ (H ∪ C)| |H|<br />
<br />
V ′ H ∪ C k − |H|<br />
<br />
V ′′ <br />
k − |H| G ′ V ′′ ∪ H k G<br />
4k<br />
G =(V, E) k <br />
M G V(M) <br />
M |V(M)| > 2k <br />
<br />
k |V − V(M)| <br />
|V − V(M)| 2k 4k <br />
|V − V(M)| >2k I = V − V(M)<br />
|N(I)| |V(M)| < |I| <br />
(C, H, R) G (C, H, R)
G ′ = G[R]<br />
k ′ = k − |H| <br />
|V − V(M)| 2k <br />
4k<br />
<br />
2k <br />
2k <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
G =(V, E) k <br />
k G E <br />
<br />
n m G N(v) <br />
v G N(v) :={u |{u, v} ∈ E} v<br />
N[v] N(v) ∪ {v}<br />
<br />
<br />
k <br />
<br />
<br />
{u, v} C <br />
u v C <br />
k
{u, v} <br />
N[u] =N[v] u <br />
u <br />
v<br />
<br />
<br />
<br />
2 k <br />
G =(V, E) 2 k <br />
C 1 ,...,C k k v ∈ V b v <br />
k i 1 i k v C i <br />
2 k u ≠ v ∈ V b u = b v b u<br />
b v u v <br />
u v
{u, v} <br />
N[u] =N[v] u <br />
u <br />
v<br />
<br />
<br />
<br />
2 k <br />
G =(V, E) 2 k <br />
C 1 ,...,C k k v ∈ V b v <br />
k i 1 i k v C i <br />
2 k u ≠ v ∈ V b u = b v b u<br />
b v u v <br />
u v
q <br />
q <br />
<br />
<br />
q = 1 <br />
<br />
(A ⊎ B) <br />
A S A<br />
<br />
q q 1<br />
q A <br />
B <br />
<br />
B q · |A|<br />
q B <br />
A<br />
<br />
|B| q·m m <br />
<br />
(n − k)
Q <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
H <br />
<br />
M H <br />
M<br />
U X X R X <br />
U X S A = X \ (U X ∪ R X ) U B <br />
B T S A M<br />
T = {x ∈ B |{u, x} ∈ M u ∈ S A } <br />
S A |B| >mq U B B<br />
H B <br />
U B A U B<br />
U X R X <br />
M U B <br />
S A S A
v ∈ A C(v) v v <br />
C(v) ∩ S A = C(v) C(v) ∩ S A = ∅ v ∈ S A<br />
v u U X {v, w} ∈ M v u <br />
{u, w} ∈ E(H) v ∈ S A v ∈ S A<br />
u R X {w, u} U X<br />
u {w, v} ∈ E(H) U X<br />
v v ∈ S A S = {v ∈ A|C(v) ⊆ S A } <br />
G[S ∪ T] q S <br />
H M v ∈ S <br />
<br />
<br />
<br />
<br />
v ∈ S A u /∈ S A u /∈ S A <br />
w v <br />
u w R X <br />
S A <br />
T S G <br />
T S A H <br />
T S G T <br />
S A H <br />
v ∈ T u ∈ N(v) u /∈ S A u ∈ R X u ∈ R X<br />
w <br />
u v {u, v} <br />
v ′ {v, v ′ } ∈ M v ′ ∈ S A <br />
w ∈ U X v ′ u ∈ U X <br />
w S <br />
T <br />
S A H
Q <br />
<br />
<br />
q <br />
<br />
<br />
<br />
(n − k) <br />
<br />
<br />
q <br />
<br />
<br />
S G \ S <br />
<br />
<br />
k k <br />
<br />
G<br />
k<br />
S ⊆ V(G) G \ S <br />
|S| k<br />
2k <br />
<br />
3k q<br />
q = 1<br />
G <br />
G M G M k <br />
G k <br />
A 2k <br />
M B = V(G) \ A G[B] <br />
B A H = (A ⊎ B, E ′ )
G A <br />
q H q = 1<br />
<br />
⎧⎪ ⎨<br />
⎪ ⎩<br />
k<br />
<br />
<br />
k <br />
<br />
M 1 H |M 1 | >k <br />
|M 1 | k |B| |M 1 | <br />
q q = 1 <br />
S ⊆ A, T ⊆ B H[S ∪ T] S <br />
H G T S <br />
G S T <br />
S ∪ T G k := k − |S|<br />
<br />
S ⊆ A, N(T) ⊆ S<br />
<br />
T ⊆ B<br />
S T
Q <br />
<br />
|B| |M| = k G[A∪<br />
B] |A ∪ B| 3k <br />
3k <br />
3k <br />
<br />
<br />
(n − k) <br />
G k (n − k) <br />
G (n − k)<br />
G (n − k)<br />
χ(G) (n − k) <br />
k <br />
k k <br />
k<br />
(n − k) <br />
G= (V, E)<br />
k<br />
V(G) (n − k) <br />
<br />
3k−3 <br />
G <br />
G G <br />
(3k − 3) q<br />
q = 1<br />
v G G <br />
G v l G <br />
G G ′ G<br />
G <br />
G χ(G) (n − k) <br />
χ(G ′ ) (n − k − l) G<br />
G G
G M ′ G M ′ k <br />
χ(G) (n − k) <br />
M ′ G <br />
|M ′ | n − 2|M ′ | G <br />
n − 2|M ′ | G <br />
n − 2|M ′ | + |M ′ | = n − |M ′ | =(n − k) <br />
|M ′ | |M| <br />
q S ⊆ A, T ⊆<br />
B; S, T ≠ ∅ H[S ∪ T] S H <br />
G T S χ(G) (n − k)<br />
χ(G \ (S ∪ T)) (n − k − |T|) <br />
G \ (S ∪ T) n ′ = n − |S| − |T| k<br />
k − |S| χ(G) (n − k) <br />
C : V → [n − k] G n − k G u T<br />
B A \ S <br />
S v ∈ S v C(u) u <br />
v M u v G <br />
u v G C(u) G<br />
(n − k) <br />
S T <br />
χ(G\(S∪T)) (n−k−|T|) <br />
<br />
S ∪ T G k := k − |S| <br />
|B| |M| k−1 G[A∪B] |A∪B| <br />
(3k − 3)<br />
(n − k) (3k − 3) <br />
k
Q <br />
<br />
<br />
<br />
e =(u, v) ∈ E(G) e ′ =(u ′ ,v ′ ) ∈ E(G) {u, v}∩<br />
{u ′ ,v ′ } ≠ φ G k <br />
G S ⊆<br />
E(G) G S <br />
k<br />
<br />
G= (V, E)<br />
k<br />
S ⊆ E(G) e ∈ E(G)<br />
S |S| k<br />
2k(k + 3) <br />
q q (k + 1)<br />
G <br />
G M G <br />
M<br />
2k G <br />
k A 4k<br />
M B = V(G) \ A<br />
G[B] B A <br />
H =(A ⊎ B, E ′ ) G <br />
A q H q =(k + 1)
⎫⎪ ⎬<br />
⎪ ⎭<br />
N(v) <br />
<br />
<br />
|N(v)| >k<br />
<br />
v<br />
v<br />
<br />
u v<br />
k <br />
<br />
<br />
<br />
M 1 H |M 1 | > 2k <br />
|M 1 | 2k |B| > |M 1 |·(k+1)<br />
q q =(k + 1) <br />
S ⊆ A, T ⊆ B H[S ∪ T] |S| (k + 1)<br />
S H G T S<br />
u ∈ S k <br />
u u <br />
u <br />
(k+1) u <br />
k <br />
u <br />
u T <br />
T S <br />
T <br />
(u, v u ) u ∈ S v u
Q <br />
<br />
<br />
S ⊆ A, N(T) ⊆ S<br />
<br />
T ⊆ B<br />
S T q q = (k + 1)<br />
k = 2<br />
|B| < |M 1 | · (k + 1) =<br />
(2k·k+1) G[A∪B] |A∪B| 2k(k+1)+4k =<br />
2k(k + 3)<br />
2k(k + 3)
Q <br />
<br />
<br />
S ⊆ A, N(T) ⊆ S<br />
<br />
T ⊆ B<br />
S T q q = (k + 1)<br />
k = 2<br />
|B| < |M 1 | · (k + 1) =<br />
(2k·k+1) G[A∪B] |A∪B| 2k(k+1)+4k =<br />
2k(k + 3)<br />
2k(k + 3)
k <br />
<br />
F k θ 2 <br />
F {θ 2 }F<br />
<br />
<br />
<br />
F k <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
G<br />
k<br />
S ⊆ V(G) G \ S G[S] <br />
|S| k
K 4 <br />
W n n <br />
n <br />
v 1 ,...,v n <br />
{v i ,v (i+1) } 1 i
(G, k) <br />
H G <br />
<br />
<br />
⋆<br />
V(H) =V(G) ∪ {v e | e ∈ E(G)} <br />
⋆<br />
E(H) ={(x, v e ), (v e ,y) | e =(x, y) ∈ E(G)}<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
(H, k) <br />
(G, k)<br />
♦ S G S<br />
H H G <br />
H<br />
♦ S H x ∈ S ⊆ V(H) v x ∈ V(G)<br />
<br />
{ x x ∈ V(G)<br />
v x =<br />
z z ∈ N(x) x H<br />
x/∈ V(G) N H (x) <br />
G x <br />
x <br />
{v x | x ∈ S} <br />
k G
⊲ <br />
(G, k)<br />
(H, l) l <br />
k H O(k 2 )<br />
⊲ ∆<br />
O(k∆)<br />
<br />
<br />
<br />
(G, k)<br />
v ∈ G <br />
<br />
X <br />
G (G \ X, k)<br />
<br />
<br />
<br />
v <br />
<br />
(k + 1)
G {x 1 ,y 1 ,z},...,{x k+1 ,y k+1 ,z} <br />
(x i ,y i ) (x i ,z) (y i ,z) 1 i (k + 1)<br />
△ i (x i ,y i ,z) <br />
((G ∪△ 1 ∪ ...∪△ k+1 ),k+ 1) (G, k) <br />
((G ∪△ 1 ∪ ... ∪△ k+1 ),k + 1) z <br />
v (v, z) <br />
v <br />
v k<br />
<br />
<br />
<br />
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(G, k) <br />
u 1 ,u 2 ,...,u r <br />
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r>(k + 2) H G <br />
{u 1 ,u 2 ,...,u r } {u a ,u b } <br />
(x, u a ), (x, u b ), (y, u a ), (y, u b ).<br />
u a u b H (H, k)<br />
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u v<br />
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→ G x x x<br />
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k = 0 G G <br />
k<br />
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G <br />
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v 1 ,...,v r <br />
(v i ,v i+1 ) i ∈ {1, 2, . . . , r − 1} v i G<br />
x y v 1 v r <br />
{v 1 ,...,v r } v x y<br />
G {v 1 ,...,v r }<br />
<br />
H<br />
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x y v i
v H <br />
x y <br />
G<br />
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H G u a u b <br />
{u a ,u b ,x,y} <br />
G x y <br />
H <br />
G k x y <br />
S k k <br />
{u 1 ,u 2 ,...,u r }.<br />
r>(k + 2) i j G \ S u i u j <br />
<br />
x − u i − y − u j − x<br />
S x y<br />
S H <br />
S x y <br />
{u 1 ,u 2 ,...,u r } x y <br />
u i x y <br />
u i <br />
x y u i S S <br />
x y <br />
r H <br />
r G r>k<br />
k <br />
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(u, v) H G u <br />
v G
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k + 2k(k),<br />
k 2k <br />
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k<br />
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G m k<br />
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H (n−t) <br />
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l<br />
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(G, k) <br />
G <br />
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v<br />
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v <br />
v (k + 1) v <br />
X ⊆ V(G) \{v} 2k v <br />
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H v v <br />
q q =(k + 2)<br />
(G, k) H v v<br />
2k+(k+2)k 2k+(k+<br />
2)k q q =(k + 2) <br />
Greduction \ H v <br />
v C 1 ,C 2 ,...,C r v <br />
v H v <br />
(G, k) k C i <br />
C 1 ,C 2 ,...,C p <br />
C p+1 ,...,C r <br />
p k <br />
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p C i H v<br />
u ∈ C i w ∈ H v (u, w) ∈ E(G) <br />
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G H v C C =<br />
{c 1 ,...,c s } c i C i <br />
s (r − p) (v, c i ) w ∈ C i<br />
{v, w} ∈ E(G)<br />
v h v H v G v<br />
C i d(v) >h v +(k + 2)h v + p <br />
(k + 2)h v <br />
q q =(k + 2) A = H v B = C S ⊆ H v<br />
T ⊆ D S |S| (k + 2) T N(T) =S<br />
(v, u) u ∈ C i <br />
c i ∈ T v w<br />
w ∈ S v w<br />
<br />
w ∈ S w v <br />
w {w a ,w b }<br />
(v, w a ), (v, w b ), (w, w a ) (w, w b ) w ∈ S <br />
w a w b c i ∈ T v <br />
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q (G, k) <br />
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k Z G G <br />
v ∈ Z S ⊆ Z<br />
w ∈ S v/∈ Z w a /∈ Z w b /∈ Z <br />
w a w b v/∈ Z w ∈ Z Z <br />
{v, w a ,w b ,w} w ∈ S<br />
v/∈ Z S ⊆ Z<br />
v ∈ Z G R \{v} G \{v} Z \{v} <br />
G R \{v} G \{v}<br />
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S v<br />
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v ∈ Z Z v Z <br />
k G S ⊆ Z <br />
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k G <br />
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G k G <br />
k v S<br />
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k <br />
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W v <br />
S x S W x 1 ,...,x k+2 <br />
x (k + 2) x<br />
|W| k i, j x i /∈ W x j /∈ W <br />
G\W {x, v, x i ,x j } <br />
W <br />
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v <br />
T W k <br />
X ⊆ W W v <br />
T v ∈ W X = ∅ W <br />
v/∈ W S ⊆ W W\X<br />
W \X k <br />
W \X k |W| k <br />
W\X W\X <br />
X S <br />
X {v} ∪ S <br />
S v <br />
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2k+(k+2)k h v +(k+2)h v +p k+(k+2)k+k<br />
h v k p <br />
p k <br />
∆ <br />
k∆ G (G, k) <br />
G n (n − t) <br />
δ ∆ G<br />
n(δ−2)−tδ <br />
2(∆−1)<br />
F G E F <br />
F G \ F n − |F| − 1<br />
G \ F <br />
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∆|F| |E F | |E(G)| − n + |F| + 1><br />
(∆ − 1)|F| ><br />
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2<br />
n(δ − 2) − tδ<br />
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2<br />
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.<br />
2(∆ − 1)<br />
− n + |F|
v <br />
T W k <br />
X ⊆ W W v <br />
T v ∈ W X = ∅ W <br />
v/∈ W S ⊆ W W\X<br />
W \X k <br />
W \X k |W| k <br />
W\X W\X <br />
X S <br />
X {v} ∪ S <br />
S v <br />
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2k+(k+2)k h v +(k+2)h v +p k+(k+2)k+k<br />
h v k p <br />
p k <br />
∆ <br />
k∆ G (G, k) <br />
G n (n − t) <br />
δ ∆ G<br />
n(δ−2)−tδ <br />
2(∆−1)<br />
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F G \ F n − |F| − 1<br />
G \ F <br />
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∆|F| |E F | |E(G)| − n + |F| + 1><br />
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G ∆ <br />
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G ∈ G H <br />
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k
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♦ H G \ H r <br />
♦ H r
|V(H)| <br />
H G \ H H <br />
+ <br />
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H <br />
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H X <br />
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H <br />
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G =(V, E) S ⊆ V ∂ G (S) S <br />
V\S S ⊆ V S N G (S) =∂ G (V\S)<br />
<br />
<br />
r G =(V, E) X ⊆ V <br />
r G |∂(X)| r (G[X]) r<br />
r X X ′ = X \ ∂(X) r <br />
X ′ X X X ′ <br />
t <br />
t t G =(V, E)<br />
t 1 t ∂(G) <br />
G ∂(G) <br />
<br />
G =(V, E) S ⊆ V <br />
G[S] |∂(S)| ∂(S) <br />
⊕ G 1 G 2 t <br />
G 1 ⊕ G 2 t G 1 G 2<br />
∂(G 1 ) ∂(G 2 ) <br />
G 1 ⊕ G 2 <br />
G 1 G 2
t ⊕
G G 1 G 2 t<br />
G 1 ,G 2 ∈ G G 1 ⊕ G 2 G <br />
G 1 ⊕ G 2 ∈ G G <br />
<br />
G =(V, E) r<br />
X X ′ X G 1 r <br />
X ′ G 1 G G[V \ X ′ ] ⊕ G 1 G[X]<br />
G 1 X ′ G 1 <br />
t <br />
Π G t<br />
G 1 G 2 G 1 ≡ Π G 2 c <br />
t G 3 k G 1 ⊕ G 3 G 2 ⊕ G 3 <br />
(G 1 ⊕ G 3 ,k) ∈ Π (G 2 ⊕ G 3 ,k+ c) ∈ Π
Π <br />
G t S t<br />
S ⊆ G t G 1 G 2 ∈ S <br />
G 2 ≡ Π G 1 S (Π,t)<br />
t ≡ Π t <br />
Π t ≡ Π <br />
<br />
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r q<br />
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r <br />
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f G <br />
f G (S) S <br />
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f G<br />
: 2 X → N <br />
f G (S) := max |I|.<br />
I G<br />
I∩X⊆S<br />
c 1 c 2 X {c} c<br />
<br />
f G <br />
G S, S ′ ⊆ X |f G (S) − f G (S ′ )| t<br />
S ⊆ X f G (∅) f G (S) f G (∅) +t
f G <br />
T = {g | g : 2 X → {0, 1, . . . , t}} t G<br />
X g ∈ T f G := 1 c + g c<br />
G \ X t <br />
G 1 G 2 ≡ T g ∈ T<br />
f G1 := 1 c1 + g f G2 := 1 c2 + g c 1 c 2 <br />
G 1 \ X G 2 \ X <br />
≡ T T <br />
2 2t+1 ≡ T <br />
Π t ≡ Π <br />
Π =<br />
≡ T ≡ Π G 1 ≡ T G 2 <br />
G 1 ≡ Π G 2 <br />
G 1 ≡ T G 2 G 1 ≡ Π G 2 <br />
t G 1 G 2 G 1 ≡ Π G 2 <br />
c t G 3 k (G 1 ⊕ G 3 ,k) ∈ Π<br />
(G 2 ⊕ G 3 ,k+ c) ∈ Π<br />
c 1 c 2 G 1 \ X G 2 \ X<br />
c 1 c 2 <br />
c := c 2 −c 1 (G 1 ⊕G 3 ,k) ∈ Π G 1 ⊕G 3 <br />
k I 1 X X ∩ I 1 <br />
G 2 I 2 g(X∩I 1 )+c 2 <br />
X X ∩ I 1 I 2 ∪ (I 1 ∩ (V(G 3 ) \ X)) <br />
G 2 ⊕ G 3 <br />
|I 2 ∪ (I 1 ∩ (V(G 3 ) \ X))| g(X ∩ I 1 )+c 2 + k − (|I 1 ∩ V(G 1 )|<br />
k + c 2 + g(X ∩ I 1 ) − (g(X ∩ I 1 )+c 1 )<br />
= k + c 2 − c 1<br />
= k + c<br />
(G 2 ⊕ G 3 ,k+ c) ∈ Π <br />
(G 2 ⊕ G 3 ,k + c) ∈ Π (G 1 ⊕ G 3 ,k) ∈ Π <br />
<br />
≡ T ≡ Π
f G <br />
T = {g | g : 2 X → {0, 1, . . . , t}} t G<br />
X g ∈ T f G := 1 c + g c<br />
G \ X t <br />
G 1 G 2 ≡ T g ∈ T<br />
f G1 := 1 c1 + g f G2 := 1 c2 + g c 1 c 2 <br />
G 1 \ X G 2 \ X <br />
≡ T T <br />
2 2t+1 ≡ T <br />
Π t ≡ Π <br />
Π =<br />
≡ T ≡ Π G 1 ≡ T G 2 <br />
G 1 ≡ Π G 2 <br />
G 1 ≡ T G 2 G 1 ≡ Π G 2 <br />
t G 1 G 2 G 1 ≡ Π G 2 <br />
c t G 3 k (G 1 ⊕ G 3 ,k) ∈ Π<br />
(G 2 ⊕ G 3 ,k+ c) ∈ Π<br />
c 1 c 2 G 1 \ X G 2 \ X<br />
c 1 c 2 <br />
c := c 2 −c 1 (G 1 ⊕G 3 ,k) ∈ Π G 1 ⊕G 3 <br />
k I 1 X X ∩ I 1 <br />
G 2 I 2 g(X∩I 1 )+c 2 <br />
X X ∩ I 1 I 2 ∪ (I 1 ∩ (V(G 3 ) \ X)) <br />
G 2 ⊕ G 3 <br />
|I 2 ∪ (I 1 ∩ (V(G 3 ) \ X))| g(X ∩ I 1 )+c 2 + k − (|I 1 ∩ V(G 1 )|<br />
k + c 2 + g(X ∩ I 1 ) − (g(X ∩ I 1 )+c 1 )<br />
= k + c 2 − c 1<br />
= k + c<br />
(G 2 ⊕ G 3 ,k+ c) ∈ Π <br />
(G 2 ⊕ G 3 ,k + c) ∈ Π (G 1 ⊕ G 3 ,k) ∈ Π <br />
<br />
≡ T ≡ Π
Π n 1 G n <br />
V(G n )={x 1 ,x 2 }∪{a 1 ,...,a n } <br />
E(G n )={(x 1 ,a 1 )}∪{(a i ,a i+1 ) | 1 i n−1} p 1 H p<br />
V(H p )={y 1 ,y 2 } ∪ {b 1 ,...,b p }<br />
E(H p )={(y 2 ,b 1 )} ∪ {(b i ,b i+1 ) | 1 i p − 1}<br />
<br />
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x 2<br />
x 1<br />
a 1 a 2 ··· a n<br />
<br />
b p ··· b 2 b 1<br />
y 2<br />
y 1<br />
<br />
G n H p <br />
x 1 y 1 <br />
x 2 y 2 G n ⊕ H p <br />
max{n, p}<br />
y 2<br />
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b p ··· b 2 b 1 x 2 x 1 a 1 a 2 ··· a n<br />
y 1<br />
<br />
G n ⊕ H p <br />
n
c k G <br />
(G n ⊕ G, k) ∈ Π (G m ⊕ G, k + c) ∈ Π <br />
c<br />
c>0 k = m G = H m G n ⊕ G <br />
m := max{n, m} (G n ⊕ G, k) ∈ Π <br />
G m ⊕G m := max{m, m} <br />
k + c (G m ⊕ G, k) /∈ Π<br />
c
t G S ⊆ V(G) <br />
(G ′ ,S ′ ) ∈ H t ζ Π G ((G′ ,S ′ )) P Π (G ⊕ G ′ ,S ∪ S ′ ) <br />
|S| ζ Π G ((G′ ,S ′ )) + f(t)<br />
<br />
<br />
+ p <br />
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<br />
<br />
<br />
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+ Π <br />
c G k r<br />
X G |X| >c O(|X|) G ∗ =(V ∗ ,E ∗ ) <br />
k ∗ |V(G ∗ )| < |V(G)| k ∗ k (G ∗ ,k ∗ ) ∈ Π <br />
(G, k) ∈ Π<br />
S (Π,2r) c = max Y∈S |Y| ζ<br />
2r 2c S <br />
2r H 2c H ≡ Π ζ(H) η <br />
2r 2c N <br />
2r H ′ k ′ <br />
(H ⊕ H ′ ,k+ η(H)) ∈ Π ⇐⇒ (ζ(H) ⊕ H ′ ,k) ∈ Π.<br />
|X| >2c 2r X ′ ⊆ X c
X ′ b b <br />
b cc H c <br />
|V(G ∗ )| < |V(G)| H k ∗ (G ∗ ,k ∗ ) ∈ Π <br />
(G, k) ∈ Π O(|X|)
X ′ b b <br />
b cc H c <br />
|V(G ∗ )| < |V(G)| H k ∗ (G ∗ ,k ∗ ) ∈ Π <br />
(G, k) ∈ Π O(|X|)
F <br />
<br />
F <br />
<br />
<br />
<br />
<br />
<br />
r r <br />
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F
⎧<br />
<br />
⎪⎨ V i V j<br />
V 1 V 2 ··· V i ··· V p<br />
|Vi| s<br />
⎪⎩<br />
U <br />
V i U<br />
⎧⎪ ⎨<br />
⎪ ⎩<br />
|N(V i ) ∩ U| (2b + 2)<br />
<br />
(s, p) b<br />
G b (s, p) G <br />
G (p + 1) V 1 ,...,V p U <br />
♦ p <br />
1 r ≠ s p u ∈ V r v ∈ V s (u, v) /∈ E(G)<br />
♦ p s |V i | s i 1 i p<br />
♦ V i U (2b + 2)<br />
i 1 i p<br />
|N(V i ) ∩ U| 2b + 2,<br />
<br />
V i U (2b + 2) G <br />
d G d · (bsp)<br />
(s, p)<br />
G b <br />
d G (d · bsp) s p G<br />
(s, p)
G (d · bsp) (X,T) <br />
G b G <br />
b <br />
<br />
<br />
T T ′ ⊆ V(T) X(T ′ ) ∪ q∈T ′X q <br />
S ⊆ V T 2p 2p <br />
T 1 ,...,T 2p T \ S i |X(T i )| 3s + b<br />
T 1 ,...,T 2p <br />
T \ S<br />
<br />
T v ∈ T C T \{v} v <br />
C v T<br />
S T <br />
r S = ∅ T r = T T r <br />
T \S r i <br />
v i V(T r ) T i<br />
T r \{v i } v i |X(T i )| 3s + b v i S T r<br />
v T r <br />
v ∈ T r C T r \ v v <br />
(3s + b) <br />
2p <br />
S T 1 ,...,T 2p <br />
2p <br />
T r = T <br />
n <br />
|X(T r )| n dbsp.<br />
T v i V(T r ) <br />
T i T r \{v i } v i |X(T i )| 3s+b<br />
T r v i T r x y <br />
T r \{x} T r \{y} (3s + b) <br />
v i T r
v i<br />
y<br />
x<br />
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⎨<br />
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⎩<br />
(3s + b)<br />
⎧⎪ ⎨<br />
⎪ ⎩<br />
2 · (3s + b)<br />
<br />
⎧<br />
v i <br />
v i <br />
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y<br />
x<br />
{ }} { { }} {<br />
((3s + b) · 2) + ((3s + b) · 2) +2,<br />
x y <br />
|X(T r )| <br />
12s + 6b 12sb<br />
2p <br />
d 32 32 (12·2)+8<br />
<br />
n>(12sb)(2p)+8(sbp)<br />
(2p − 1) <br />
n>(12sb) · 1 + 8(sb)p,
G<br />
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<br />
G<br />
S ′ S = {x, y, z}<br />
4(sb) (3s+b) <br />
<br />
<br />
S T 1 ,...,T 2p S ′ <br />
S S ′ = S u v <br />
S ′ t V T t S ′ <br />
l l <br />
|S ′ | 2|S| v <br />
v <br />
<br />
S S<br />
S <br />
S |S|<br />
S ′ <br />
|S ′ | 2|S| S ∗ = S ′ \ S<br />
|S| >2p |S ′ | 2|S| <br />
|S ∗ | 2p.<br />
p T 1 ,...,T 2p <br />
S ∗ p T 1 ,...,T 2p <br />
S ∗ T 1 ,...,T p S ∗<br />
i p T i S ∗ T ′<br />
i = T i
T \ S ′ |X(T i ′)|<br />
s T i v S ∗ v <br />
3 |X(T i )| 3s + b T i \{v} T i ′ |X(T i ′ )| s<br />
V i = X(T i ′)<br />
U V i |V i | s<br />
T i ′ S ′ u v<br />
u = v T i ′ S ′ N(V i ) ⊆ X u ∪ X v <br />
|N(V i )| 2b + 2 <br />
f <br />
f : V(G) → N<br />
ˆf : 2V → N <br />
ˆf(S) = ∑ v∈S<br />
f(v).<br />
(f, s, p) <br />
G b ˆf <br />
(f, s, p) G G (p + 1)<br />
V 1 ,...,V p U <br />
♦ p <br />
1 r ≠ s p u ∈ V r v ∈ V s (u, v) /∈ E(G)<br />
♦ p ˆf(Vi ) s i 1 i p<br />
♦ V i U (2b + 2)<br />
i 1 i p<br />
|N(V i ) ∩ U| 2b + 2,<br />
<br />
<br />
G b <br />
f : V(G) → N ˆf : 2V → N <br />
ˆf(S) = ∑ v∈S<br />
f(v).<br />
d G (d·bsp) <br />
s p G (f, s, p)
(2τ + 1) X τ <br />
X G τ <br />
X <br />
<br />
X <br />
<br />
G[X] X<br />
<br />
<br />
⋆ X <br />
⋆ ∂(X)<br />
<br />
<br />
∂(X) <br />
<br />
<br />
→ n G X (G[X]) τ<br />
← 2(τ + 1) G <br />
( )<br />
|X|<br />
.<br />
4|∂(X)| + 1<br />
G[X] <br />
τ τ <br />
G[X] (T, B = {B l } l∈V(T ) ) T <br />
T <br />
v ∈ ∂(X) l T v ∈ B l <br />
|∂(X)| T <br />
u v <br />
T M T
G \ X<br />
<br />
<br />
<br />
∂(X) ={x, y, z}<br />
<br />
<br />
x<br />
<br />
z<br />
X<br />
<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
X<br />
<br />
l l <br />
|M| 2|∂(X)| v <br />
v <br />
<br />
<br />
<br />
∂(X) <br />
<br />
|M| 2|∂(X)|<br />
T T \ M 2|M| + 1 <br />
C 1 ,C 2 ...C η η <br />
2|M| + 1<br />
i C i M <br />
<br />
M C i <br />
T r i A i B i <br />
r i A i<br />
B i M B i A i <br />
M <br />
r i <br />
T \ M A i B i <br />
A i <br />
C i <br />
<br />
C i
• C i <br />
T C i<br />
C i <br />
<br />
• C i <br />
r i C i <br />
r i <br />
M C i<br />
<br />
C i <br />
C i <br />
i η D i C i M <br />
C i <br />
D i := C i ∪ N T (C i ).<br />
P i G D i <br />
P i = ⋃<br />
u∈D i<br />
B u .<br />
P i <br />
P i ∂(X) <br />
∂(X) C i N T (C i )<br />
<br />
A B <br />
A B v ∂(X) B v<br />
A C i
C i A v<br />
v <br />
∂(X) C i N T (C i ) <br />
P i ∩ ∂(X) N T (C i ) <br />
(τ+1) C i 2(τ+1) ∂(X)<br />
P i 2(τ + 1) G<br />
η 2|M|+1 4|∂(X)|+1 <br />
|X|<br />
P i M P 4|∂(X)|+1 1 ...P η<br />
P i <br />
<br />
<br />
θ c<br />
θ c <br />
F F <br />
θ c <br />
⋆ θ c <br />
⋆ θ c <br />
<br />
⋆ <br />
θ c <br />
<br />
⋆ Θ c <br />
T v k O(1) <br />
v v/∈ T v <br />
<br />
θ c <br />
θ c <br />
<br />
θ c <br />
c ∈ N M G <br />
θ c G M T 1 T 2 S c <br />
T 1 T 2
θ c <br />
θ 9 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
c 2 θ c M G <br />
G M 2 <br />
M M<br />
M <br />
x M<br />
x M T 1 S <br />
x x T 1 θ c <br />
M M <br />
x M T 1 T 1 1 ,T2 1 ,...,Tl 1<br />
T 1 x T 1 M ′ <br />
x M l>0 T i 1 <br />
T 2 l = 0 M ′ = T 2 M ′ <br />
x <br />
<br />
<br />
<br />
<br />
φ <br />
t <br />
G t <br />
S G φ f(t, |φ|)|V(G)|
θ c <br />
<br />
θ c <br />
θ c <br />
<br />
θ c <br />
M θ c <br />
v v <br />
<br />
H G φ H (G) <br />
G H G<br />
H V(H) ={h 1 ,...,h c } φ H (G) <br />
φ H (G) ≡∃X 1 ,...,X c ⊆ V(G)[<br />
∧<br />
i≠j<br />
(X i ∩ X j = ∅) ∧ ∧<br />
∧<br />
(h i ,h j )∈E(H)<br />
1ic<br />
Conn(G, X i )∧<br />
∃x ∈ X i ∧ y ∈ X j [(x, y) ∈ E(G)]<br />
] <br />
conn(G, X) G[X]<br />
G c <br />
X 1 ,...,X c (h i ,h j ) ∈ E(H) <br />
G X i X j <br />
X 1 ,...X c x 1 ,...x c G[x 1 ,...,x c ] H<br />
H G φ H (G) <br />
G H φ H (G)<br />
<br />
F F F<br />
G <br />
S ⊆ V(G)[ ∧ H∈F<br />
¬φ H (G \ S)]
S G\S H<br />
H ∈ F S <br />
F <br />
θ c <br />
θ c <br />
G v ∈ V(G) θ c l<br />
v l θ c G v <br />
v<br />
v θ c <br />
v M θ c <br />
G v v<br />
<br />
S ⊆ V(G) :<br />
∃F ⊆ E(G)<br />
⎡ ⎛<br />
⎧⎪ ⎨ ⎢<br />
⎣ ∀x ∈ S ⎜<br />
⎝ ∃X ⊆ V ′ ⎪ ⎩<br />
(G ′ ,X)<br />
∧<br />
x ∈ X<br />
∧ ∀y ∈ S[y ≠ x =⇒ y/∈ X]<br />
∧ φ θc (X ∪ {v})<br />
⎫⎞⎤<br />
⎪⎬<br />
⎟⎥<br />
⎠⎦<br />
⎪⎭<br />
<br />
S S<br />
θ c <br />
v F <br />
G ′ V(G) F<br />
V ′ = V(G) \{v} x ∈ S <br />
X G x X ∪ {v} <br />
θ c <br />
S <br />
G ′ x, y ∈ S X Y <br />
X
Y u ∈ X u ∈ Y u<br />
y Y G ′ y ∈ X X <br />
G ′ <br />
S <br />
<br />
F θ c <br />
θ c <br />
{θ c } F <br />
<br />
G n v G <br />
t ∈ O(1) G O(n) <br />
S ⊆ V G G \ S θ c <br />
{M 1 ,M 2 ,...,M l } θ c G <br />
1 i
F<br />
20 2l5 <br />
(l×l) S F G k (l×<br />
l) H (G \ S) 20 2l5 (G) <br />
k + d d = 20 2l5 (k + d) <br />
S <br />
G \ S <br />
<br />
F = θ c <br />
<br />
θ c <br />
θ c <br />
(2c − 1)<br />
<br />
F θ c G F S <br />
k (G) k +(2c − 1)<br />
S F G k θ c ∈ F <br />
(G \ S) (2c − 1) (G) k +(2c − 1) <br />
(k + d) <br />
S G\S
F<br />
20 2l5 <br />
(l×l) S F G k (l×<br />
l) H (G \ S) 20 2l5 (G) <br />
k + d d = 20 2l5 (k + d) <br />
S <br />
G \ S <br />
<br />
F = θ c <br />
<br />
θ c <br />
θ c <br />
(2c − 1)<br />
<br />
F θ c G F S <br />
k (G) k +(2c − 1)<br />
S F G k θ c ∈ F <br />
(G \ S) (2c − 1) (G) k +(2c − 1) <br />
(k + d) <br />
S G\S
F P <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
F <br />
<br />
<br />
O(OPT 2√ log OPT) <br />
O(log 3/2 OPT)<br />
<br />
{F} <br />
F <br />
H H ∈ F F <br />
<br />
<br />
<br />
<br />
S G \ S <br />
<br />
G F <br />
S G \ S H <br />
H ∈ F F <br />
<br />
F =<br />
{K 2,3 ,K 4 } F = {K 3,3 ,K 5 } F = {K 3 ,T 2 }
F P <br />
<br />
K i,j i j<br />
K i i T 2 <br />
<br />
F <br />
F <br />
F G F <br />
F S ⊆ V(G) G \ S H <br />
H ∈ F<br />
F <br />
F<br />
<br />
<br />
<br />
G F<br />
k<br />
F S |S| k<br />
F <br />
F <br />
<br />
F <br />
<br />
F <br />
<br />
F F <br />
<br />
<br />
F<br />
F<br />
G F <br />
k<br />
F S |S| k
F <br />
<br />
H F <br />
h |H|<br />
G H<br />
H <br />
<br />
S G H <br />
G \ S <br />
S <br />
G H S <br />
<br />
<br />
<br />
F <br />
OPT <br />
F <br />
⊲ OPT <br />
H <br />
S<br />
G H <br />
⊲ (OPT · |S|) <br />
<br />
H<br />
<br />
<br />
H <br />
<br />
H
F P <br />
♦ H h (t × t) t =(14h −<br />
24) <br />
♦ 20 2t5 t×t <br />
h H (G) ><br />
20 2(14h−24)5 G H <br />
F <br />
d (20 2(14h−24)5 + 1) <br />
d<br />
G H G H <br />
<br />
F <br />
G H (d + 1)<br />
F <br />
<br />
<br />
G H <br />
G G <br />
F G <br />
k <br />
<br />
<br />
F h G <br />
F S k (G \ S) d (G) k + d d =<br />
20 2(14h−24)5 <br />
<br />
<br />
F<br />
k <br />
<br />
n <br />
k
(G) d F G <br />
d <br />
l <br />
<br />
(G) l d ′ (G) √ log (G),<br />
d ′ l > (k + d)d ′√ log(k + d) <br />
F G (k + 1) <br />
<br />
G <br />
(G) l (k + d)d ′√ log(k + d).<br />
S G H <br />
(d + 1) <br />
<br />
<br />
(T, X = {X t } t∈V(T ) ) G <br />
H t <br />
t <br />
t <br />
H t := G t [V(G t ) \ X t ].<br />
β : V(T) → N <br />
β(t) =(H t ).<br />
β <br />
β <br />
<br />
• t H t (H t ) <br />
• t s t H t H s <br />
β <br />
• t s t H t <br />
H s H s H t
F P <br />
<br />
t : {b, x, y}<br />
<br />
{b, x, y}<br />
{b, x, y}<br />
<br />
{b, x}<br />
{b, y}<br />
<br />
<br />
{b}<br />
{b}<br />
{a, b}<br />
β(t) =(G[{a, c}])<br />
{a, b}<br />
{b}<br />
<br />
{a, b}<br />
{a, b, c}<br />
{a, b}<br />
<br />
<br />
<br />
<br />
{a}<br />
<br />
β() <br />
• t r s H t <br />
<br />
max{(H s ), (H r )},<br />
r s β <br />
<br />
β <br />
<br />
β() <br />
<br />
β
β(t) =(H t ) >d <br />
t β(t) =(H t ) >d<br />
P = V(H t ) Q =(V(G) \ P) \ X t S = X t X t <br />
t G H G ∗ <br />
P Q G H <br />
d H <br />
(d + 1) t <br />
β(t) >d β() <br />
F G H <br />
G H <br />
G ∗ S <br />
d
F P <br />
(G, k)<br />
(G) d <br />
S F G <br />
|S| >k<br />
<br />
<br />
S<br />
<br />
<br />
(T, X = {X t } t∈V(T ) ) l<br />
l > (k + d) √ log(k + d) d <br />
G F k<br />
<br />
(T, X = {X t } t∈V(T ) ) <br />
V(G) G H G ∗ X t <br />
T (G H )=(d + 1) <br />
<br />
G H Y <br />
Z (G ∗ ,k− (G H ))<br />
Z |Y| >k <br />
<br />
<br />
X ∪ Y ⋃ (G ∗ ,k− (G H ))
F <br />
(G, k) F <br />
O(k 2√ log k) <br />
<br />
<br />
k G S <br />
<br />
S F G<br />
|S| = O(k 2√ log k) <br />
S <br />
l<br />
(k + d) √ log(k + d) d <br />
<br />
<br />
<br />
<br />
k <br />
G k <br />
G k <br />
<br />
<br />
<br />
G ∗ <br />
G ∗ k G ∗ <br />
G G k <br />
<br />
<br />
G F k<br />
d <br />
d
F P <br />
(G, k)<br />
α <br />
(G) α · (k + d)<br />
G H S<br />
(G) > α · (k + d)<br />
<br />
⋆ d
F <br />
d (d + 1) <br />
<br />
<br />
S(G H ) ∪ X t<br />
⋃ ( (G ∗ ) ) ,<br />
• G H <br />
• S(G H ) F G H <br />
• X t G H <br />
• G ∗ (G \ G H ) \ X t <br />
• |X t | (k + d)d ′√ log(k + d) X t <br />
<br />
M H H G <br />
• M H G H <br />
• M H G ∗ <br />
• M H X t M H X t G H G ∗ <br />
M H <br />
X t <br />
M H <br />
• S(G H ) G H <br />
• (G ∗ ) G ∗ d <br />
F G ∗ <br />
<br />
• X t <br />
F G <br />
O(k 2√ log k) S(G, k)<br />
(G, k) G ∗ i G i H <br />
G ∗ G H i th X i t<br />
i th <br />
<br />
S(G ∗ i,i)=OPT(G (i−1)<br />
H<br />
)+|S(G ∗ (i−1),i− 1)| + |X i t|.
F P <br />
G ∗ 0 H<br />
S(G ∗ 0 ,0) <br />
S(G) <br />
|S(G ∗ 0,0)| = 0.<br />
OPT(G k H)+|S(G ∗ k,k− 1)| + |X k t |.<br />
<br />
S(G) =(OPT(G k H)+OPT(G k−1<br />
H<br />
)+···+ OPT(G1 H)) + k · max{X i t}.<br />
i<br />
OPT(G i H ) >k 1 i k <br />
OPT(G i H ) k Xi t = O(k √ log k) 1 i <br />
k <br />
<br />
S(G) =k 2 + k · O(k √ log k),<br />
S(G) =O(k 2√ log k).<br />
<br />
<br />
k <br />
<br />
k G H G ∗ X t <br />
k G H <br />
<br />
G H <br />
<br />
G H G ∗ X t <br />
d
n<br />
<br />
<br />
<br />
<br />
<br />
<br />
k 1 k n 1 <br />
k G F k − 1<br />
OPT k S <br />
O(k 2√ log k) <br />
F OPT <br />
F G <br />
S ⊆ V(G) G[V \ S] F |S| =<br />
O(OPT 2 · √OPT)<br />
<br />
<br />
O(OPT ·√log<br />
OPT)<br />
O(log OPT 3/2 ) <br />
<br />
<br />
<br />
<br />
(G, k) <br />
F<br />
µ <br />
⊲ µ k<br />
⊲ X G A<br />
G B µ(G A ) µ(G B ) <br />
µ(G)
F P <br />
⊲ µ(G) <br />
G A G B X <br />
<br />
<br />
<br />
<br />
λ <br />
λ · ,<br />
<br />
<br />
k <br />
<br />
log(k O(1) ) O(log k) <br />
<br />
λ · O(log k).<br />
<br />
O(k √ log k) <br />
O(OPT · √log<br />
OPT) O(log OPT 3/2 )<br />
<br />
O(k 2√ log k) <br />
<br />
<br />
<br />
m m =O(k 2√ log k) <br />
µ(G) G<br />
µ(G) := G.
G <br />
<br />
(k + d) √ log(k + d) <br />
<br />
<br />
<br />
<br />
<br />
<br />
t µ(H t ) H t <br />
<br />
t t <br />
µ H t <br />
<br />
⊲ µ(H t ) t µ(H s ) s <br />
t<br />
⊲ µ(H t ) t µ(H s ) s <br />
t v <br />
H t H s <br />
⊲ µ(H t ) t µ(H r )+µ(H s ) r s <br />
t<br />
µ(H t ) <br />
(m/α) t α µ(H t ) > (m/α)<br />
µ(H s ) (m/α) s t t <br />
t <br />
µ() <br />
<br />
t s H s G \ (H t ∪ X t ) <br />
X s µ(G \ H t ) m(1 −<br />
1/α) m <br />
H t (m/α) t µ(G \ (H t ∪<br />
X t )) (m(1 − 1/α)) G \ (H t ∪ X t ) G \ H t <br />
α 1
F P <br />
t r s <br />
t X t = X s = X r G A G B<br />
<br />
X t <br />
µ(H s ) m/α µ(H r ) m/α,<br />
µ(H s )+µ(H r ) m<br />
µ(H t ) > m/α,<br />
<br />
µ(G \ (H t ∪ X t )) (1 − 1/α)m.<br />
µ(H s ) µ(H r ) <br />
m/2α G A G B <br />
µ(H s ) m/2α G A H s ∪ G \ (H t ∪ X t ) G B H r <br />
µ(G B ) m/α <br />
µ(G A ) m/2α +m(1 − 1/α) =(1/2α + 1 − 1/α)m =<br />
(2α − 1)<br />
2α<br />
· m,<br />
m α<br />
µ(H s ) µ(H r ) m/2α <br />
<br />
µ(H s ) > m/2α µ(H r ) > m/2α.<br />
µ(H s )=m(1/2α + δ s ) µ(H r )=m(1/2α + δ r ).<br />
µ(H s )+µ(H r ) m <br />
(1/α + δ s + δ r 1) ⇒ δ s + δ r 1 − 1/α,
δ s δ r (1 − 1/α)/2 <br />
δ s α − 1<br />
2α ,<br />
H s ∪ G \ (H t ∪ X t ) H r <br />
µ(G B ) m/α µ(H s ∪ G \ (H t ∪ X t ))<br />
µ(H s ∪ G \ (H t ∪ X t )) m(1/2α +(α − 1)/2α)+m(1 − 1/α),<br />
<br />
µ(H s ∪ G \ (H t ∪ X t )) m ·<br />
(3α − 2)<br />
.<br />
2α<br />
m (3α−2)
F P <br />
d ′ l > (k + d)d ′√ log(k + d) <br />
F G (k + 1)<br />
<br />
G <br />
(G) l (k + d)d ′√ log(k + d).<br />
G A G B X <br />
<br />
<br />
(T, X = {X t } t∈V(T ) ) G H t <br />
<br />
t t <br />
H t := G t [V(G t ) \ X t ].<br />
β : V(T) → N <br />
β(t) =|H t ∩ S|.<br />
β <br />
β <br />
µ() <br />
β <br />
t <br />
β(t) > (2/3)m G A G B G <br />
<br />
• t s G A H s G B<br />
G \ (H t ∪ X t )<br />
• t s r<br />
• β(s) m/2 G A H s ∪ G \ (H t ∪ X t ) G B H r <br />
• β(r) m/2 G A H r ∪ G \ (H t ∪ X t ) G B H s <br />
<br />
G A G B X <br />
G A ∩Z G B ∩Z <br />
k <br />
k k
t : {b, x, y}<br />
<br />
{b, x, y}<br />
{b, x, y}<br />
<br />
{b, x}<br />
{b, y}<br />
<br />
<br />
{b}<br />
{b}<br />
{a, b}<br />
β(t) =G[{a, c}] ∩ Z<br />
{a, b}<br />
{b}<br />
<br />
{a, b}<br />
{a, b, c}<br />
{a, b}<br />
<br />
<br />
<br />
<br />
{a}<br />
<br />
β()
F P <br />
<br />
β <br />
<br />
<br />
<br />
<br />
F <br />
<br />
<br />
O(k(log k) 3/2 ) <br />
<br />
k F G <br />
O(k(log k) 3/2 )<br />
<br />
G j (1),G j (2),...,G j (2 j )<br />
i <br />
2 i <br />
i <br />
0 G G A<br />
G B G <br />
<br />
j <br />
λ j (1), λ j (2),...,λ j (2 j )<br />
2 j <br />
λ j λ j <br />
j<br />
<br />
λ j = λ j (1)+λ j (2)+...+ λ j (2 j ).
k j (1),k j (2),...,k j (2 j )<br />
F G j (1),G j (2),...,G j (2 j )<br />
k F G <br />
<br />
G <br />
<br />
<br />
j (j + 1) <br />
j <br />
<br />
k j (1)+k j (2)+...+ k j (2 j ) k.<br />
G j (i) 1 i 2 j <br />
(k j (i)+d) <br />
<br />
√<br />
(k j (i)+d) log(k j (i)+d).<br />
λ j (i) <br />
<br />
√<br />
λ j (i) =(k j (i)+d) log(k j (i)+d),<br />
λ j <br />
∑<br />
√<br />
(k j (i)+d) log(k j (i)+d).<br />
1i2 j<br />
(k + d(2 j )) √ log(k + d)
F P <br />
k j (1)+k j (2)+...+ k j (2 j ) k<br />
(k j (1)+d)+(k j (2)+d)+...+(k j (2 j )+d) k + d(2 j )<br />
(<br />
(kj (1)+d)+(k j (2)+d)+...+(k j (2 j )+d) ) √ log(k + d) (k + d(2 j )) √ (<br />
log(k + d)<br />
∑ ) √log(k<br />
1i2<br />
(k j j (i)+d) + d) (k + d(2 j )) √ log(k + d)<br />
∑<br />
(<br />
∑1i2<br />
(k j j (i)+d) √ )<br />
log(k + d)<br />
(k + d(2 j )) √ log(k + d)<br />
1i2 j (<br />
(kj (i)+d) √ log(k j (i)+d) ) (k + d(2 j )) √ log(k + d),<br />
k j (i) k <br />
λ j (k + d(2 j )) √ log(k + d).<br />
j <br />
(k + d(2 j )) √ log(k + d) <br />
<br />
S(G, S) (G, S) <br />
<br />
∑<br />
1i2<br />
|S(G j j (i),S j (i))| = ∑ 1i2<br />
λ j j +<br />
∑<br />
|S((G<br />
∑ 1i2j j (i)) A ,S j (i) ∩ (G j (i)) A )| +<br />
1i2<br />
|S((G j j (i)) B ,S j (i) ∩ (G j (i)) B )|<br />
(G j (i)) A G j+1 (2i − 1) (G j (i)) B<br />
G j+1 (2i) <br />
(j + 1) <br />
j S ∩ G A <br />
|S(G j (i), ∅)| = 0<br />
<br />
<br />
|S(G 1 (1), (S ∩ G 1 (1)))| + |S(G 1 (2), (S ∩ G 1 (2)))| + λ 0 ,
|S(G A , (S ∩ G A ))| + |S(G B , (S ∩ G B ))| + λ 0 .<br />
γ <br />
<br />
<br />
S(G) =λ 0 + λ 1 + ...+ λ γ .<br />
λ j (k + d(2 j )) √ log(k + d),<br />
<br />
(k √ log(k + d)) · γ + ∑<br />
1jγ<br />
d(2 j )<br />
<br />
G A G B <br />
<br />
|S ∩ G A | 5 6 |S|, |S ∩ G A| 1 3 |S|<br />
|S ∩ G B | 2 3 |S|<br />
O(log |S|) O(log(k 2√ k)) <br />
O(log k) γ <br />
k <br />
∑<br />
1jγ<br />
d(2 j )<br />
(k √ log(k + d)) · γ
F P <br />
(k √ log(k + d)) · (log k) <br />
O(k(log k) 3/2 ) <br />
<br />
k 1 k n 1 <br />
k G F k−1 <br />
OPT k S <br />
O(OPT(log OPT) 3/2 ) <br />
<br />
F OPT <br />
F G <br />
S ⊆ V(G) G[V \ S] F |S| =<br />
O(OPT · (OPT) 3/2 )<br />
(G, S)<br />
S ∩ G = ∅ <br />
∅<br />
<br />
(T, X = {X t } t∈V(T ) ) l<br />
(T, X = {X t } t∈V(T ) ) <br />
V(G) G A G B X t <br />
T |G A ∩ S| (5/6)|S| |G B | (2/3)|S| <br />
<br />
(G A ,S∩ G A ) ∪ (G B ,S∩ G B ) ∪ X t<br />
F<br />
<br />
F <br />
η <br />
η η<br />
G <br />
η<br />
1
η <br />
O(log 3/2 n)
F P
{F} <br />
<br />
<br />
<br />
<br />
<br />
<br />
F <br />
F K 1,t <br />
<br />
F F<br />
K 1,t <br />
t <br />
K 1,t <br />
(t−1) K 1,t K 1,3 <br />
<br />
K 1,7 <br />
<br />
<br />
K 1,t <br />
O(k log k) <br />
K 1,t <br />
coNP ⊆ NP/poly<br />
<br />
k <br />
F h G F<br />
S k (G \ S) d (G) k + d d =<br />
20 2(14h−24)5
{F} <br />
<br />
<br />
<br />
<br />
<br />
r<br />
r F <br />
r <br />
+ Π <br />
γ : N → N (G, k)<br />
r X G γ(r) O(|X|) <br />
(G ∗ ,k ∗ ) |V(G ∗ )| < |V(G)| k ∗ k (G ∗ ,k ∗ ) ∈ Π <br />
(G, k) ∈ Π<br />
G K 1,t <br />
G ′ K 1,t <br />
<br />
K 1,t <br />
G <br />
G G G ′ <br />
G<br />
r<br />
<br />
F <br />
<br />
<br />
<br />
<br />
<br />
<br />
K 1,t <br />
F
♦ X G τ τ <br />
X (2τ + 2) X <br />
X <br />
<br />
(<br />
)<br />
|X|<br />
.<br />
4 · |∂(X)| + 1<br />
X S<br />
F S G<br />
<br />
G \ S X<br />
<br />
α<br />
γ(α) <br />
<br />
γ(α) <br />
<br />
<br />
<br />
{(<br />
}} ){<br />
|X|<br />
γ(2τ + 2)<br />
4 · |∂(X)| + 1<br />
|X| γ(2τ + 2)+4 · |∂(X)| + 1.<br />
X ∂(X)<br />
<br />
X <br />
X G \ S (|X| + |S|) |S|<br />
k <br />
∂(X) S <br />
♦ F h G K 1,t <br />
S F ∂ G (G\S) ≡ N(S) g(h, t)·|S| <br />
g h t
{F} <br />
⎫<br />
S<br />
} F <br />
<br />
|S| = (k)<br />
X<br />
(X) =d<br />
⎪⎬<br />
<br />
<br />
G<br />
(|X|/q) <br />
<br />
⎪⎭<br />
(<br />
γ(2d + 2) ><br />
X<br />
q<br />
|X| qγ(2d + 2)<br />
)<br />
<br />
<br />
(q = 4N(S)+1)<br />
N(S) |S| · <br />
(G ∗ ,k ∗ )<br />
|V| = |X| + |S| <br />
<br />
<br />
<br />
F <br />
<br />
<br />
<br />
<br />
X<br />
<br />
<br />
G K 1,t F <br />
h S F X<br />
G \ S <br />
g h t<br />
|∂(X)| g(h, t) · |S|,
X X S <br />
S X<br />
∂(X) = ⋃ v∈S<br />
N X (v).<br />
<br />
|∂(X)|<br />
∑ v∈S |N X(v)|<br />
|S| · max v∈S |N X (v)|.<br />
<br />
v ∈ S h t<br />
X = G \ S (X) d d = 20 2(14h−24)5 <br />
d d+1 <br />
|N X (v)| (t − 1)(d + 1) <br />
G[N X (v)] (G \ S) d (G[N X (v)]) d <br />
G[N X (v)] (d + 1) <br />
κ G[N X (v)] d + 1 <br />
t <br />
t K 1,t G<br />
(d + 1) t <br />
<br />
|N X (v)| (t − 1)(d + 1).<br />
<br />
g(h, t) =(t − 1)(20 2(14h−24)5 + 1),<br />
|∂(X) |S| · g(h, t) <br />
<br />
<br />
F <br />
<br />
+ p
{F} <br />
F <br />
<br />
<br />
F <br />
F <br />
p <br />
F Π = F<br />
t G ∂(G) <br />
S ′′ ⊆ V(G) G G \ S ′′ <br />
F S = S ′′ ∪ ∂(G)<br />
(G ′ ,S ′ ) ∈ H t ζ Π G ((G′ ,S ′ )) G ⊕<br />
G ′ [(V(G) ∪ V(G ′ )) \ (S ∪ S ′ )] F |S| <br />
ζ Π G ((G′ ,S ′ )) + t F <br />
F <br />
<br />
<br />
<br />
<br />
(G, k) F h <br />
F <br />
F O(log 3/2 OPT)<br />
S G \ S F <br />
S O(k log 3/2 k) (G, k) <br />
F
X V(G) \ S d <br />
G[X] d 20 2(14h−24)5 <br />
|X|<br />
2(d + 1) Y G F<br />
4|∂(X)|+1<br />
γ : N → N <br />
|X|<br />
γ(2d+1) 2(d+1)<br />
4|∂(X)|+1<br />
Y G (G ∗ ,k ∗ ) |V(G ∗ )| < |V(G)| k ∗ <br />
k (G ∗ ,k ∗ ) F (G, k) <br />
F G ∗ K 1,t <br />
<br />
(G ∗ ,k ∗ ) X (2d+<br />
2) γ(2d + 2) G ∗ \ X <br />
O(k log 3/2 k) <br />
<br />
k ∗ k <br />
|V(G ∗ )| γ(2d + 2)(4|∂(X)| + 1)+|S|.<br />
|∂(X)| g(h, t) · |S| <br />
|X|<br />
4|∂(X)|+1<br />
|V(G ∗ )| = O(γ(2d + 2) · k log 3/2 k)=O(k log 3/2 k).<br />
γ(2d + 2)<br />
O(k log 3/2 k)<br />
(2d + 2)<br />
(2d + 2) Y <br />
γ(2d + 2) ∂(Y) 2d + 2 <br />
k O(d) (G ∗ ,k ∗ ) <br />
G F X <br />
k (G\X) d X <br />
|V(G ∗ )| = O(k) G ∗ <br />
O(k) <br />
G <br />
<br />
<br />
K 1,t <br />
F<br />
O(k log k) K 1,t
{F} <br />
G k <br />
k G <br />
<br />
<br />
l G l <br />
S ⊆ V(G) (4l log l) G \ S <br />
G k 2 <br />
S G \ S S <br />
(8k log k) G k <br />
+ <br />
G \ S (G ∗ ,k ∗ ) <br />
<br />
(G \ S) 1 (G, k) <br />
V(G ∗ ) O(k log k)<br />
O(k log k) <br />
K 1,t
Θ c <br />
<br />
<br />
<br />
<br />
<br />
θ c c 1 <br />
<br />
F θ c F<br />
O(k 2 log 3/2 k)<br />
F <br />
<br />
Θ c <br />
F<br />
θ c ∈ F <br />
θ c <br />
S <br />
G \ S <br />
θ c <br />
<br />
θ c c = 7<br />
Θ c <br />
c = 1 <br />
c = 2 <br />
<br />
c = 3 <br />
c = 2 O(k 2 )
Θ C <br />
O(k 2−ε ) coNP ⊆ NP/poly <br />
<br />
<br />
F <br />
<br />
<br />
<br />
k <br />
k O(1) <br />
F <br />
θ c <br />
<br />
q <br />
<br />
<br />
<br />
<br />
v ∈ V(G) T v<br />
k O(1) θ c v v <br />
θ c k <br />
θ c v/∈ T v <br />
G \{v} <br />
θ c θ c G v <br />
<br />
<br />
θ c <br />
v<br />
G \{v}<br />
G n θ c v <br />
G ′ = G \{v} θ c <br />
v k T v O(k) v/∈ T v<br />
G \ T v θ c T v
θ c (2c−1) <br />
G ′ (2c − 1)<br />
T v <br />
β <br />
<br />
l <br />
<br />
(G) l d ′ (G) √ log (G),<br />
d ′ (G ′ ) (2c − 1) =O(1) <br />
<br />
<br />
(T, X = {X t } t∈V(T ) )<br />
G H t <br />
t <br />
t <br />
H t := G t [V(G t ) \ X t ].<br />
M θc (G, v) θ c <br />
G v M θc (G, v) <br />
θ c v<br />
β : V(T) → N <br />
β(t) =M θc (H t ∪ {v},v),<br />
β t θ c v <br />
H t ∪ {v} G ′ <br />
v<br />
β <br />
β <br />
<br />
• t H t (H t )
Θ C <br />
• t s t H t H s <br />
β <br />
• t s t H t <br />
H s θ c v (H s ∪<br />
{v}) (H t ∪ {v})<br />
• t r s <br />
θ c H t <br />
M θc (H s ∪ {v},v)+M θc (H r ∪ {v},v),<br />
β() <br />
<br />
β() <br />
β() t θ c <br />
v H t ∪ {v} β() <br />
t <br />
β() β() <br />
t β(r) =0 t <br />
<br />
G ∗ V(G ′ ) \ H t S <br />
s s t <br />
t s <br />
G ∗ S <br />
G ′ ∪ {v} θ c <br />
<br />
θ c v <br />
θ c v k <br />
k <br />
β() <br />
<br />
<br />
θ c v
t : {b, x, y}<br />
<br />
{b, x, y}<br />
{b, x, y}<br />
<br />
{b, x}<br />
{b, y}<br />
<br />
<br />
{b}<br />
{b}<br />
{a, b}<br />
β(t) =M θc (G[{a, c} ∪ {v}])<br />
{a, b}<br />
{b}<br />
<br />
{a, b}<br />
{a, b, c}<br />
{a, b}<br />
<br />
<br />
<br />
<br />
{a}<br />
<br />
β()
Θ C <br />
<br />
<br />
θ c v G ∗ <br />
θ c v H t ∪ {v} S <br />
(H t ∪ {v}) \ S θ c <br />
H s ∪ {v} <br />
θ c β(s) t <br />
β() <br />
k s 1 ,...,s k <br />
1, . . . , k <br />
G ′ <br />
<br />
G ′ G ′ <br />
<br />
|X si | = O(1),<br />
X s s <br />
k · (G ′ )=O(k)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
k<br />
(G, k) Θ c <br />
(G ′ ,k ′ ) k ′ k <br />
G ′ k <br />
Θ c
v<br />
<br />
<br />
θ 5 k = 3 <br />
v k 2<br />
<br />
<br />
v θ c <br />
(G, k)<br />
G<br />
G v ∈ V(G) θ c l <br />
v l θ c G v <br />
v <br />
θ c k <br />
v G v G <br />
(G \{v}, (k − 1))<br />
<br />
v <br />
θ c <br />
θ c <br />
k <br />
θ c k
Θ C <br />
(G, k) Θ c (G, k) <br />
G <br />
k G θ c k θ c <br />
O(k log 3/2 k) G <br />
θ c k (G, k) Θ c <br />
S O(k log 3/2 k) <br />
<br />
H v θ c <br />
v v/∈ H v <br />
S θ c v ∈ S<br />
S v := S \{v} G v := G \ S v <br />
(2c−1) θ c v<br />
G v θ c (G v ) (2c−1)+1 = O(1) <br />
v G v <br />
v ∈ S G v k+1 <br />
(G ← G \{v},k← k − 1) <br />
(G, k)<br />
<br />
v ∈ S <br />
v G v k <br />
v ∈ V(G) H v ⊆ V(G)\{v} <br />
θ c v v<br />
H v {v} v/∈ S H v = S <br />
v ∈ S v G v <br />
k T v O(k) <br />
v θ c v G v <br />
H v = S v ∪ T v |H v | h v H v <br />
V(G) \{v} <br />
v H v <br />
<br />
q q = c<br />
(G, k) S H v v<br />
ch v +c(c−1)h v ch v +<br />
c(c − 1)h v q q = c
Gv<br />
⎧⎪ ⎨<br />
⎪ ⎩<br />
<br />
v<br />
G \ S<br />
(Gv) =O(1)<br />
<br />
⎫⎪ ⎬<br />
⎪ ⎭<br />
<br />
<br />
G<br />
Sv S \ v S θc <br />
<br />
v<br />
<br />
(G ∗ ,k ∗ )<br />
<br />
k Gv \{v} θc<br />
Hv ← Tv ∪ Sv<br />
Tv <br />
θc v Gv v/∈ Tv<br />
Tv ∪ Sv <br />
v G<br />
Hv k O(1) <br />
v v/∈ Hv Hv
Θ C <br />
<br />
v H v θ c G H v <br />
θ c G<br />
v ch v + c(c − 1)h v <br />
G \ H v v<br />
C 1 ,C 2 ,...,C r v (c − 1) <br />
θ c <br />
H v θ c C i <br />
θ c <br />
C i H v u ∈ C i w ∈<br />
H v (u, w) ∈ E(G) <br />
H v G <br />
θ c u C<br />
H v G[V(C) ∪ {v}] θ c <br />
u M ⊆ G v ∈ M <br />
u ′ ∈ M u ′ /∈ C ∪ {v} v<br />
u u ′ M c 2 <br />
θ c M G <br />
<br />
D 1 ,D 2 ,...,D s i D i <br />
H v <br />
G <br />
<br />
H v v ∈ V(G) w
v<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
H v<br />
⋆ <br />
⋆ <br />
c c = 5<br />
G H v D D =<br />
{d 1 ,...,d s } d i D i <br />
(v, d i ) w ∈ D i (v, w) ∈ E(G)<br />
<br />
<br />
<br />
c v ch v H v v (c−1)<br />
D i d(v) >ch v + c(c − 1)h v <br />
|D| ch v q q =<br />
c A = H v B = D S ⊆ H v T ⊆ D S |S| c<br />
T N(T) =S <br />
(v, u) u ∈ D i <br />
d i ∈ T c v w w ∈ S <br />
<br />
c v w c <br />
v w c <br />
v <br />
ch v + c(c − 1)h v ,<br />
h v := |H v | H v θ c θ c v<br />
G H v D D = {d 1 ,...,d s }
Θ C <br />
v<br />
<br />
<br />
T ⊆ D<br />
<br />
c c = 5<br />
S ⊆ H v ,N(T) ⊆ S<br />
<br />
v<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
⋆<br />
H v<br />
⋆ <br />
<br />
v T ⋆ S<br />
T
v <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
⋆<br />
H v<br />
<br />
⋆ <br />
T <br />
<br />
d i D i G \ H v x ∈ H v<br />
d i ∈ D D i <br />
x<br />
q q = c A = H v B = D S ⊆ H v<br />
T ⊆ D S |S| c T N(T) =S<br />
♦ (v, u) u ∈ D i d i ∈ T <br />
♦ v w w ∈ S <br />
v w c v w (c − r) <br />
r <br />
q q = c G R<br />
<br />
<br />
G S v G R <br />
c (G, k) Θ c (G R ,k)<br />
Θ c <br />
G R Z k <br />
Z θ c G v ∈ Z S ⊆ Z<br />
v ∈ Z G R \{v} G \{v} Z \{v}
Θ C <br />
G R \{v} G \{v} Z <br />
k G S ⊆ Z <br />
k G k<br />
G R k <br />
k v S <br />
W v S <br />
|S| c G[S ∪ T] v θ c <br />
v G v <br />
D D i <br />
d i ∈ T X W<br />
D i ∈ D W ′ W <br />
X S W ′ := (W \ X) ∪ S<br />
W ′ k S ′ <br />
S W W<br />
W D θ c <br />
c X ′ ⊆ X <br />
S ′ X ′ |W ′ | |W| k<br />
W ′ θ c G <br />
M u ∈ X u ∈ D i i<br />
M H v \ S v u H v \ S G \ S<br />
M <br />
<br />
q <br />
S S G R <br />
<br />
<br />
<br />
<br />
(G, k)<br />
Θ c (G ′ ,k ′ ) k ′ k <br />
G ′ O(k log 3/2 k)<br />
(H, l) <br />
(H, l)
(G, k)<br />
(G ∗ ,k ∗ ) <br />
<br />
(G ∗ ,k ∗ ) ≡ (G, k) <br />
<br />
<br />
(G ∗ ,k ∗ )<br />
<br />
<br />
S O(k log 3/2 k)<br />
v ∈ S v θ c k <br />
(G \ S) ∪ {v} <br />
v θ c k <br />
(G ∗ ,k ∗ ) <br />
(G ∗ ,k ∗ )<br />
<br />
T v <br />
H v := T v ∪ S v S v := S \{v}<br />
<br />
v/∈ S H v := S<br />
[G, k, {H v | v ∈ V(G)]<br />
(G, k) H<br />
ch v + c(c − 1)h v O(k log 3/2 k) <br />
<br />
v ∈ V(G) λ(v) v v c<br />
q <br />
λ(v) λ(v) <br />
λ(v) <br />
<br />
⎛<br />
⎝ ∑<br />
⎞<br />
λ(v) ⎠ + n<br />
v∈V(G)<br />
<br />
∑<br />
v∈V(G)<br />
λ(v) n 2 ,
Θ C <br />
(G, k)<br />
X := (G, k)<br />
G, k H v v ∈ V(G) X<br />
v ∈ V(G) <br />
d(v) >ch v + c(c − 1)h v <br />
(G ∗ ,k ∗ ) <br />
(G ∗ ,k ∗ )<br />
<br />
<br />
(G, k)<br />
<br />
<br />
<br />
<br />
Θ c <br />
O(k log 3/2 k)<br />
(G, k) Θ c <br />
(G ′ ,k ′ ) k ′ k <br />
G ′ O(k log 3/2 k) θ c <br />
S G ′ O(k log 3/2 k) d <br />
S d := (G \ S)<br />
<br />
<br />
<br />
<br />
<br />
θ c (2c − 1) d 2c − 1 <br />
2(d + 1) Y G ′ |V(G′ )|−|S|<br />
<br />
4|N(S)|+1<br />
Θ c γ : N → N <br />
|V(G′ )|−|S|<br />
γ(2d + 1) <br />
4|N(S)|+1<br />
2(d + 1) Y G ′ G ∗
|V(G ∗ )| < |V(G ′ )| k ∗ k ′ (G ∗ ,k ∗ ) Θ c <br />
(G ′ ,k ′ ) Θ c <br />
<br />
<br />
<br />
<br />
G <br />
<br />
<br />
<br />
(G ∗ ,k ∗ ) S <br />
(2d + 2) γ(2d + 2) G ∗ \ S <br />
<br />
O(k 2 log 3 k) <br />
|V(G∗ )|−|S|<br />
γ(2d + 2) 4|N(S)|+1<br />
k∗ k <br />
|V(G ∗ )| γ(2d + 2)(4|N(S)| + 1)+|S|<br />
<br />
γ(2d + 2)(4|S|∆(G ∗ )+1)+|S|<br />
γ(2d + 2)(O(k log 3/2 k) × O(k log 3/2 k)+1)+O(k log 3/2 k)<br />
<br />
O(k 2 log 3 k).<br />
(G ∗ \<br />
S) (2c − 1) =d G ∗ \ S d|V(G ∗ ) \<br />
S| = O(k 2 log 3 k). S <br />
|S| · ∆(G ∗ ) O(k 2 (log k) 3 ) <br />
O(k 2 log 3 k)<br />
<br />
<br />
(2d + 2) <br />
2d + 2 Y γ(2d + 2) ∂(Y)<br />
2d + 2 k O(d) (G ∗ ,k ∗ ) <br />
<br />
∆(G ∗ )=O(k log 3/2 k) G θ c S <br />
k (G \ S) (2c − 1) =d <br />
S |V(G ∗ )| = O(k 2 log 3/2 k) |E(G ∗ )| O(k 2 log 3/2 k)<br />
G ∗
Θ C <br />
O(k 2 log 3/2 k) G <br />
<br />
<br />
<br />
O(k 2 log 3/2 k)<br />
<br />
<br />
<br />
2k O(k 2 )
F<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
F <br />
<br />
F F <br />
<br />
• S (k + 1) G <br />
• G \ S G[S] F <br />
• F k <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
+ <br />
G V(G) A B G[A] G[B]<br />
S ⊆ B k G \ S <br />
F <br />
<br />
<br />
F
F<br />
G \ S<br />
<br />
S<br />
G \ S G[S] F<br />
F k G \ S <br />
F <br />
<br />
F<br />
G F k F S<br />
(k + 1)<br />
k<br />
F X |X| k S ∩ S = ∅<br />
F<br />
F O(k 3 ) <br />
F <br />
<br />
<br />
F 2 O(k log k) n 2 <br />
F
F<br />
<br />
G[B] <br />
G \ B <br />
F <br />
<br />
<br />
<br />
F <br />
<br />
<br />
F <br />
<br />
<br />
F<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
H V(H) E(H) <br />
V(H) <br />
2 H I(H)<br />
V(H) ∪ E(H) v ∈ V(H) e ∈ E(H) I(H)<br />
v ∈ e
F<br />
<br />
{a, b, c, d, e, f, g}<br />
{a, b, e, g}{b, e}{d, e, f}{b, g} {c, g}<br />
s h <br />
K h s h <br />
s h n K h <br />
(s h<br />
√<br />
log h)n <br />
s h = 0.319 . . . + o(1) <br />
G K h <br />
G 2 s h√<br />
log h<br />
n <br />
<br />
<br />
H n <br />
I(H) K h |E(H)| 2 s h√<br />
log h<br />
n<br />
<br />
H <br />
e H e <br />
H e ′ ⊂ e H <br />
H e e ′ <br />
I(H) e<br />
H 2 H<br />
G G V(H) G <br />
H <br />
G H G I(H)
G <br />
I(H) 2 s h√<br />
log h<br />
n<br />
H n <br />
I(H) K h <br />
H h 2 s h√<br />
log h<br />
h(h − 1)n/2<br />
h h(h −<br />
1)/2 I(H) K h <br />
<br />
(s, p) <br />
<br />
<br />
G b (s, p) G <br />
G (p + 1) V 1 ,...,V p U <br />
♦ p <br />
1 r ≠ s p u ∈ V r v ∈ V s (u, v) /∈ E(G)<br />
♦ p s |V i | s i 1 i p<br />
♦ V i U (2b + 2)<br />
i 1 i p<br />
|N(V i ) ∩ U| 2b + 2,<br />
G b d G<br />
(d · bsp) s p G (s, p)<br />
<br />
<br />
<br />
<br />
b d<br />
G (d · bsp) s p G <br />
(s, p) (s, p)
F<br />
s p <br />
s p<br />
k <br />
k O(1) <br />
<br />
(G, S, k) F <br />
c α β F <br />
G (α(k + 1) +k) · c · (β(k + 1) +k) <br />
r c r <br />
F<br />
G \ S G \ S <br />
b (s, p)<br />
G \ S s p <br />
h <br />
F F H <br />
(V 1 ,...,V p ,U) (s, p) G \ S<br />
α <br />
p (s, p) (G\S) (α·(k+1)+k) <br />
V i V i h <br />
S<br />
C 1 ,...,C γ G[V \ (U ∪ S)] <br />
H S C i 1 i γ <br />
e i = N(C i ) ∩ S<br />
I(H) G \ U <br />
C i <br />
G \ U G \ U H I(H) H<br />
<br />
G \ U H G <br />
F S ∗ ⊆ (V \ S) k<br />
G \ S ∗ H <br />
H ∗ I(H ∗ ) H<br />
H ∗ k H <br />
I(H ∗ ) H ∗ (k + 1)
h H ∗ <br />
<br />
2 h√ s log h<br />
h(h − 1)<br />
· (k + 1).<br />
2<br />
α 2 s h√<br />
log h<br />
h(h − 1)/2<br />
V i <br />
h S <br />
H h S<br />
h H ∗ <br />
H H ∗ <br />
<br />
(α · (k + 1)) <br />
k <br />
p (α·(k+1)+k) <br />
h S<br />
S<br />
<br />
<br />
<br />
S <br />
β s <br />
(s, p) G \ S β · k <br />
<br />
H ∗ H <br />
(k + 1) <br />
H <br />
2 s h√<br />
log h<br />
· (k + 1).<br />
H k H ∗ <br />
H <br />
β 2 s h√<br />
log h<br />
<br />
2 s h√<br />
log h<br />
· (k + 1)+k.
F<br />
e h H c[e] <br />
C N(C) =e w(c[e]) <br />
C N(C) =e <br />
β(k + 1)+k <br />
β(k+1)+k<br />
∑<br />
i=1<br />
w(c[e i ]) = s.<br />
s>(β(k + 1) +k)c <br />
c e<br />
e <br />
<br />
<br />
<br />
<br />
F<br />
+ <br />
<br />
<br />
G 1 G 2 t <br />
i ∈ {1, 2} f Gi V(G i ) <br />
[t] v ∈ G i f Gi <br />
G 1 G 2 G 1<br />
∼ =t G 2 <br />
h : V(G 1 ) → V(G 2 ) h (u, v) ∈ E(G 1 ) <br />
(h(u),h(v)) ∈ E(G 2 ) f G1 (v) =f G2 (h(v)) h <br />
<br />
G 1 G 2<br />
G t G t <br />
1 t t G <br />
µ G : V(G) → 2 [t] µ G <br />
v l ∈ [t] {l} µ G (v) ={l} <br />
v ∈ (V(G) \ ∂(G)) µ G (v) =∅
H <br />
f H : V(H) → 2 [t] (u, v) ∈ E(H) H ′ <br />
H u v w uv <br />
(u, v)<br />
H H ′ f H ′ : V(H ′ ) →<br />
2 [t] x ∈ V(H ′ ) ∩ V(H) f H ′(x) =f H (x) w uv <br />
f H ′(w uv ) = f H (u) ∪ f H (v) <br />
t <br />
H f : V(H) → 2 [t] G <br />
t µ G H <br />
G H G <br />
<br />
<br />
G<br />
H t <br />
G <br />
H<br />
h t <br />
h t G M h (G) t<br />
G h <br />
t h G 1 G 2 t<br />
G 1 ≡ h t G 2 M h (G 1 )=M h (G 2 )<br />
t h t h <br />
M h (G) <br />
h t ≡ h t <br />
t <br />
≡ h t <br />
h t h <br />
h <br />
h 2 (h 2) · (2 t ) h h <br />
2 2(h 2)+th ≡ h t
F<br />
F t G 1<br />
G 2 G 1 ≡ F G 2 t G 3 Z ⊆ V(G 3 )<br />
(G 1 ⊕ G 3 \ Z) F (G 2 ⊕ G 3 \ Z) F .<br />
<br />
F q = max H∈F {|V(H)|} <br />
t l =(q + t) ≡ l t ≡ F <br />
G 1 ≡ l t G 2 G 1 ≡ F G 2 t ≡ F <br />
G 1 ≡ l t G 2 G 1 ≢ F G 2 <br />
t G 3 Z ⊆ G 3 <br />
(G 1 ⊕ G 3 \ Z) F (G 2 ⊕ G 3 \ Z) F <br />
(G 1 ⊕ G 3 \ Z) F (G 2 ⊕ G 3 \ Z) F <br />
G 1 ,G 2 G 3 <br />
(G 2 ⊕ G 3 \ Z) F <br />
H ∈ F (G 2 ⊕ G 3 \ Z) M (2,3)<br />
H<br />
<br />
φ : V(H) →<br />
2 V(G′ )<br />
v ∈ V(H) G ′ [φ(v)] <br />
v, u ∈ V(H) φ(u)∩φ(v) =∅ uv ∈ E(H) <br />
u ′ v ′ ∈ E(G ′ ) u ′ ∈ φ(u) v ′ ∈ φ(v) P 1 ,P 2 ,...,P h <br />
M (2,3)<br />
H Q i P i ∩ V(G 2 )<br />
Q i P i ∂(G 2 ) <br />
G 2 <br />
t <br />
∪ h i=1 Q i (h + t)<br />
H ∗ <br />
∪ h i=1 Q i G 2 H ∗<br />
l G 2 G 1 G 2 <br />
H ∗ l G 1 G 1 <br />
H ∗ <br />
R 1 ,...,R x X ∪ h i=1 Q i
∂(G 2 ) R j G 1 <br />
<br />
R 1 ,...,R x <br />
M (1,3)<br />
H<br />
H G 1 ⊕ G 3 \ Z Y i P i ∩ V(G 3 ) <br />
M (1,3)<br />
H<br />
Y i R j <br />
∂(G 3 ) {P i | P i ∩ ∂(G 3 ) ≠ ∅} <br />
M (2,3)<br />
H<br />
<br />
X X B X ∂(G 3 ) <br />
X B R 1 ,...,R x <br />
i X 1 ,...,iX m <br />
∪ m j=1R i X<br />
j<br />
∩ ∂(G 3 )=X B .<br />
M X ∪ m j=1 R i X ∩ V(G X<br />
j<br />
3 )<br />
M X G 1 ⊕ G 3 \ Z <br />
X B M X <br />
M X \ X B X B <br />
X G 2 ⊕G 3 \Z <br />
(X ∩ V(G 2 )) <br />
T 1 ,...,T m <br />
{R i X<br />
j<br />
} m i=1 T i ∂(G 2 ) <br />
t i T i ∩ ∂(G 2 ) X <br />
G 2 ⊕ G 3 \ Z P ij t i t j 1 i ≠ j m <br />
P ij (X∩V(G 3 )) <br />
T 1 ,...,T m T x G 2 <br />
T i T j G 2 <br />
(V(G 3 ) ∩ X) \ ∂(G 3 ) T x P ij <br />
P ij <br />
T x a b <br />
a b R i X x<br />
<br />
T x (G 1 ⊕G 3 \Z) {a, b} ⊆ R i X x<br />
<br />
R i X x<br />
a b <br />
(G 1 ⊕ G 3 \ Z) Q ij t i <br />
t j G 1 ⊕ G 3 \ Z M X <br />
X M X M (1,3)<br />
H<br />
M(1,3)<br />
H<br />
<br />
H G 1 ⊕ G 3 \ Z
F<br />
G 1 ⊕G 3 \Z F <br />
≡ F ≡ h t <br />
≡ F t<br />
<br />
F c <br />
G r X G |X| >c O(|X|) <br />
G ∗ |V(G ∗ )| < |V(G)| Z ⊆ (V(G) \ X) ∪ ∂(X)<br />
(G \ Z) F (G ∗ \ Z) F .<br />
S ≡ F c =<br />
max Y∈S |Y| ζ 2r 2c <br />
S 2r H 2c H ≡ Π ζ(H)<br />
|X| >2c 2r X ′ ⊆ X c
(G, Y, k) <br />
P Π (G, T) T G k T ⊆ Y <br />
F (G, G \ S, k)<br />
G[S] G\S F <br />
P Π (G, T) (∧ H∈F ¬φ H (G)) φ H (G) <br />
H G <br />
<br />
O(k 3 )<br />
(G, S, k) F <br />
p Y = G \ S <br />
(G, S, k) G[S] G \ S <br />
F Y (G ′ ,S,k)<br />
Y ′ ⊆ G ′ \ S G F<br />
Z ⊆ Y k G ′ F Z ′ ⊆ Y ′ <br />
k G ′ [S] G ′ \ S F <br />
V(G) <br />
<br />
(G, S, k) G[S] G \ S F Y<br />
α β<br />
F G <br />
c·(α(k+1)+k)·(β(k+1)+k) r c <br />
r <br />
F<br />
G c · (α(k + 1) +<br />
k) · (β(k + 1) +k) c <br />
X G <br />
F p + 1 <br />
q O(|X|) Z ⊆ X ∩ Y<br />
|Z| qk W ⊆ Y W (G, S, k)<br />
W ′ W ′ X<br />
Z W ′ ∩ X ⊆ Z (G, S, k)<br />
Y (G, S, k) (Y \ X) ∪ Z<br />
+ 2 Q X O(k) <br />
O(k) X 1 ,...,X l X<br />
∪ l i=1 X i = X Q
F<br />
Q ∩ X i ⊆ ∂(X i ) Q Z ∪ (S ∩ X)<br />
c qk + dl <br />
X i X γ d <br />
X γ ∩ Z ∂(X γ ) <br />
<br />
X γ <br />
X γ \ ∂(X γ ) G <br />
X γ F G X γ <br />
G ′ G ′ <br />
(G ′ ,S,k) Y <br />
G ′ \S F G\S <br />
F O(n) <br />
X γ <br />
<br />
<br />
<br />
<br />
(G ′ ,S,k)<br />
Y ′ ⊆ G ′ \ S |V(G)| = O(k 3 ) G ′ [S] G ′ \ S<br />
F<br />
<br />
O(k 3 )<br />
O(k 3 ) <br />
O(n 2 ) <br />
F<br />
p<br />
F O(k 3 )<br />
<br />
<br />
<br />
F
n G m <br />
G F 11<br />
+ G k G<br />
F k F S l =<br />
ηk log 3/2 k G F <br />
k S v 1 ,...,v l V i =<br />
{v 1 ,...,v i } ∪ (V(G) \ S) 1 i l G F Y<br />
k Y ∩ V i F G[V i ] i l Y <br />
F G[V i ] Y ∪ {v i+1 } F G[V i+1 ] <br />
1 l i th G[V i ] S k + 1 G[V i ]<br />
k <br />
S <br />
k Z G[S \ Z] F <br />
(G, S \ Z, k − |Z|) F <br />
F 2 O(k log k) n 2 <br />
2 k+1 <br />
F G[V i ] k <br />
G F k F Y <br />
k G[V i ] Y G[V i+1 ],Y∪ v i+1 ,k)<br />
F 2 k+1<br />
l <br />
F<br />
(G, S, k) F Y <br />
G \ S <br />
(G ′ ,S,k ′ ) Y ′ <br />
G F k G\S G ′ <br />
F k ′ k (G ′ \S)∩Y ′ |V(G ′ )| dk 3 <br />
k (G ′ \S)∩Y ′<br />
( )<br />
dk 3<br />
k 2<br />
O(k log k)<br />
<br />
F <br />
F O((2 k+1 2 k log k )2 k+1 l) S <br />
O(n 2 )<br />
11 + <br />
<br />
O(2 O(k) n 2 ) F<br />
2 O(k log k) n 2 11 +
F<br />
<br />
O(n + m) =O(n 2 ) <br />
O(2 O(k) n log n) (G) >k <br />
5k 11<br />
+ <br />
S O(2 O(k) n 2 )
M θc<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
G H <br />
<br />
♦ H G <br />
H H<br />
♦ H S ⊆ V(G) H <br />
G \ S H<br />
<br />
<br />
<br />
H <br />
f : N → N k 0 G <br />
H k H f (k)<br />
H H = MH <br />
H H = θ 1 H = θ 2 H = MH <br />
<br />
<br />
<br />
♦ θ 1 <br />
k<br />
2k
M θC<br />
♦ H = Mθ 2 <br />
f(k) =O(k log k) <br />
k <br />
O(k log k)<br />
H = Mθ c <br />
c>0 <br />
Mθ c F F = θ c <br />
θ c G S ⊆ V(G) <br />
S θ c G \ S θ c <br />
<br />
θ c <br />
<br />
θ c G k <br />
θ c θ c f(k) =O(k 2 log k)<br />
<br />
θ c<br />
Mθ c <br />
G 2c 2 k 2 G k<br />
θ c <br />
G 2c 2 k 2 G k <br />
θ c G θ c ηk 2 = O(k 2 )<br />
η c<br />
<br />
G k θ c <br />
G 2c 2 k 2 <br />
G θ c f(k)
V(G) <br />
<br />
B G <br />
<br />
<br />
(l × l)<br />
<br />
<br />
⋆ <br />
S <br />
<br />
<br />
⋆ B B <br />
<br />
<br />
⋆ G G<br />
<br />
G<br />
<br />
(l × l) <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
B <br />
G (T, {T v : v ∈ V(T)}) G <br />
B B B T B :=<br />
∪{T v : v ∈ V(B)} T <br />
{T B : B ∈ B} <br />
T B i B j v <br />
x ∈ T v ∈ T x G<br />
<br />
x T Bi T Bj (u, v)
M θC<br />
u ∈ B i v ∈ B j y ∈ T (u, v) ∈ T y <br />
G <br />
y T Bi T Bj <br />
{T B : B ∈ B} T<br />
<br />
x T x<br />
V B B |T x | <br />
G B<br />
<br />
(1 + (G)) <br />
<br />
<br />
G k <br />
(k + 1)<br />
<br />
B G G <br />
B<br />
P G B <br />
v P X<br />
P v v P<br />
P Z X Z <br />
Q v X Z Q ∩ P = {v} P ∪ Q<br />
Z P B<br />
θ c <br />
c <br />
k <br />
θ c
G 2c 2 k 2 <br />
G k θ c <br />
G 2c 2 k 2 G <br />
B 2c 2 k 2 + 1 <br />
P v 1 ,...,v t <br />
P P t 2c 2 k 2 +1<br />
P B B<br />
1 i t B i B v i <br />
1 i t ∪ i j=1 B j O i <br />
s O s c 2 k 2 s<br />
O 1 = 1 O t >2c 2 k 2 1 i t − 1 O i+1 <br />
O i + 1<br />
B 1 = ∪ s i=1 B i B 2 = B \ B 1 O i <br />
B 2 c 2 k 2 <br />
B 1 B 2 B<br />
B P 1 P <br />
v 1 v s P 2 v s+1 v t <br />
P 1 P 2 c 2 k 2 <br />
P c 2 k 2 <br />
P 1 P 2 P 1 P 2<br />
S c 2 k 2 S <br />
B 1 B 2 c 2 k 2 <br />
A ∈ B 1 ,B ∈ B 2 A ∩ S = ∅ = B ∩ S A B <br />
B A ∩ P 1 ≠ ∅,B∩ P 2 ≠ ∅ S P 1 P 2 <br />
P∪P 1 ∪P 2 k θ c E p <br />
P 1 P 2 P i ∈<br />
{1, 2} Q i = P i ∩ E p Q 1 Q 2 [M]<br />
M = |Q 1 | = |Q 2 | f :[M] → [M] f(i) =j
M θC<br />
v 1<br />
v 2 ··· v s v s+1 ··· v t−1 v t<br />
O 1 O 2 ··· O s O s+1 ··· O t−1 O t<br />
|O <br />
s | =(c 2 k 2 + 1)<br />
v 1 ∈ B i1<br />
v t ∈ B j1<br />
v 1 ∈ B i2<br />
v t ∈ B jt<br />
v 1 ∈ B i3<br />
<br />
···<br />
···<br />
<br />
v t ∈ B j3<br />
<br />
<br />
<br />
v 1 ∈ B ip<br />
v t ∈ B jp<br />
B 1<br />
B 1 B 2 <br />
B 2
P i j <br />
C ⊆ P i, i ′ ∈ Q 1 ∩C; if(i ′ )<br />
P f(1),f(2),...,f(M)<br />
M <br />
〈f(1),f(2),...,f(M)〉 <br />
√ |M| = ck i ∈ {1, 2} Q ′ i Q i<br />
P <br />
Q ′ 1 ,Q′ 2 P 1 ,P 2 <br />
k θ c <br />
<br />
<br />
<br />
G 2c 2 k 2 <br />
k ′ = k − 1 θ c S ⊆ V(G) |S| =<br />
O(k 2 ) G \ S θ c <br />
<br />
<br />
<br />
<br />
<br />
<br />
G 2c 2 k 2 k ′ = k − 1 <br />
θ c O(k 2 ) <br />
<br />
µ <br />
<br />
⊲ µ k<br />
⊲ X G A<br />
G B G A G B <br />
<br />
⊲
M θC<br />
G A G B<br />
X X <br />
G A G B <br />
<br />
µ <br />
<br />
O(k 2 )<br />
µ<br />
θ c G <br />
µ <br />
θ c <br />
G <br />
µ(G) <br />
µ k O(1) <br />
G (k − 1) <br />
θ c <br />
<br />
<br />
<br />
G <br />
t µ <br />
H t <br />
t <br />
t m µ(G)<br />
µ H t <br />
<br />
⊲ µ(H t ) t µ(H s ) s <br />
t<br />
⊲ µ(H t ) t µ(H s ) s <br />
t v <br />
H t H s
⊲ µ(H t ) t µ(H r )+µ(H s ) r s <br />
t<br />
µ(H t ) <br />
(m/3) t α µ(H t ) > (m/3) µ(H s ) <br />
(m/3) s t t <br />
t <br />
µ() <br />
<br />
t s H s G \ (H t ∪ X t ) <br />
X s µ(G\H t ) m(1− 1 3 )<br />
m H t<br />
(m/3) t µ(G \ (H t ∪ X t )) (m(1 − 1 3 ))<br />
G \ (H t ∪ X t ) G \ H t <br />
t r s t<br />
X t = X s = X r G A G B <br />
<br />
X t <br />
µ(H s ) m/3 µ(H r ) m/3,<br />
µ(H s )+µ(H r ) m<br />
µ(H t ) > m/3,<br />
<br />
µ(G \ (H t ∪ X t )) <br />
(<br />
1 − 1 )<br />
m.<br />
3<br />
µ(H s ∪ H r ) (2/3)m µ(G \ (H t ∪ X t )) (2/3)m <br />
G A := (H s ∪ H r ) G B :=<br />
G \ (H t ∪ X t ) <br />
G S G \ S <br />
G A G B <br />
µ(G A ) (2/3)µ(G) µ(G B ) (2/3)µ(G).
M θC<br />
<br />
2c 2 k 2 <br />
<br />
<br />
<br />
<br />
(k/2) 2c 2 (k/2) 2 <br />
<br />
<br />
<br />
µ <br />
<br />
θ c d <br />
20 2t5<br />
t×t (d+1)c×d <br />
d θ c 20 2t5<br />
t = dc <br />
θ c d <br />
θ c d <br />
20 (2dc)5 <br />
µ(G) =1 <br />
<br />
<br />
t <br />
0 1 X t <br />
t θ c <br />
G\X H t H 1 ,...,H p H p+1 ,...,H q <br />
G \ X θ c H i 1 i p <br />
t µ() 0 1<br />
θ c H i p
µ(G) =1 G θ c <br />
<br />
θ c <br />
θ c <br />
<br />
(G, µ(G))<br />
(T, X = {X t } t∈V(T ) ) G<br />
(T, X = {X t } t∈V(T ) ) <br />
µ(G) =1 <br />
<br />
<br />
V(G) G A G B X t <br />
T µ(G A ) (2/3)µ(G) µ(G B ) (2/3)µ(G)<br />
<br />
(G A , (2/3)µ(G)) ∪ (G B , (2/3)µ(G)) ∪ X t<br />
<br />
<br />
<br />
<br />
<br />
O(k 2 )<br />
θ c <br />
<br />
<br />
t G <br />
X t <br />
θ c G A G B <br />
S 1 S 2 <br />
G \{S 1 ∪ S 2 } <br />
G A G B X t <br />
X t G A G B S 1 ∪ S 2 ∪ X t <br />
θ c G
M θC<br />
<br />
<br />
S(G, µ(G)) S(G A , (2/3)µ(G)) + S(G B , (2/3)µ(G)) + (G).<br />
(G) c 2 µ(G) 2 µ(G) <br />
k k µ(G) <br />
S(G, µ(G)) T(G, k) <br />
T(G, k) =T(G A ,2/3k)+T(G B ,2/3k)+c 2 k 2 .<br />
k = 1 <br />
T(G, 1) =O(1).<br />
T(G, 1) =O(1) <br />
<br />
T(G, k) =c 2 k 2 +<br />
( d∑<br />
i=1<br />
( 8<br />
9) i<br />
· c 2 k 2 )<br />
+ O(1)<br />
i → ∞<br />
<br />
<br />
d∑<br />
) i<br />
.<br />
i=1<br />
( 8<br />
9<br />
O(k 2 ) <br />
<br />
{θ c } <br />
<br />
k k <br />
θ c G
{θ C } <br />
<br />
θ c <br />
G k<br />
k<br />
S 1 ,S 2 ,...,S r ⊆ V r k G[S i ] θ c<br />
E(S i ) ∩ E(S j )=∅ 1 i r<br />
<br />
<br />
<br />
θ c <br />
<br />
v H v v<br />
v <br />
v <br />
<br />
<br />
θ c <br />
G k θ c θ c <br />
f(k) =O(k 2 )<br />
θ c <br />
<br />
θ c O(k 2 ) θ c O(k 2 log k)<br />
<br />
θ c <br />
θ c O(k 2 log k)<br />
S <br />
S <br />
v <br />
θ c G v<br />
r r v r <br />
V(G)<br />
{S 1 ,S 2 ,...,S r }
M θC<br />
v<br />
S i ∩ S j = {v}, ∀1 i, j r<br />
G[S i ] θ c 1 i r<br />
k v G <br />
<br />
r<br />
r r <br />
θ c k <br />
k θ c <br />
<br />
v <br />
θ c <br />
<br />
θ c k <br />
θ c <br />
k <br />
v ∈ S S v := S \{v} G v := G \ S v <br />
(2c − 1) θ c <br />
v G v θ c c (G v ) <br />
(2c − 1) +1 = O(1) <br />
v G v v ∈ S G v<br />
k + 1 <br />
v ∈ S v G v <br />
k<br />
v ∈ V(G) H v ⊆ V(G)\{v} <br />
θ c v v H v<br />
{v} v /∈ S H v = S <br />
v ∈ S v G v <br />
k T v O(k) <br />
v θ c v G v <br />
H v = S v ∪ T v |H v | h v H v
{θ C } <br />
<br />
V(G) \{v} <br />
v H v <br />
v H v θ c G H v<br />
θ c G G \ H v <br />
v C 1 ,C 2 ,...,C r v <br />
(c − 1) <br />
θ c H v θ c <br />
C i θ c <br />
C i H v u ∈ C i w ∈<br />
H v (u, w) ∈ E(G) <br />
H v G <br />
θ c u C<br />
H v G[V(C) ∪ {v}] θ c <br />
u M ⊆ G v ∈ M <br />
u ′ ∈ M u ′ /∈ C ∪ {v} v<br />
u u ′ M c 2 <br />
θ c M G <br />
<br />
D 1 ,D 2 ,...,D s i D i <br />
H v <br />
G <br />
<br />
H v v ∈ V(G) w <br />
<br />
G H v D D =<br />
{d 1 ,...,d s } d i D i <br />
(v, d i ) w ∈ D i (v, w) ∈ E(G)<br />
k<br />
θ c v v <br />
H v <br />
{u 1 ,...,u hv }<br />
H v G
M θC<br />
u i x i <br />
u i u i <br />
u j j
{θ C } <br />
<br />
<br />
k O(1) <br />
<br />
<br />
<br />
θ c
M θC
⊆ / <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Q, Q ′ ⊆ {0, 1} ∗ Q Q ′<br />
D x =(x 1 ,...,x t )<br />
x i ∈ {0, 1} ∗ i ∈ [t] D(x) ∈ {0, 1} ∗ <br />
|D(x)| =(max i∈[t] |x i |) O(1)<br />
f <br />
f(k)p(n) <br />
k
D(x) ∈ Q ′ i ∈ [t] :x i ∈ Q<br />
Q ′ = Q Q <br />
Q Q ′ Q ′ <br />
<br />
NP <br />
⊆ /<br />
<br />
<br />
<br />
<br />
<br />
Π <br />
Π A x =<br />
((x 1 ,k), (x 2 ,k),...,(x t ,k)) x i ∈ {0, 1} ∗ i ∈ [t] <br />
A(x) :=(y, l) ∈ {0, 1} ∗ × N <br />
l = k O(1)<br />
(y, l) ∈ Π i ∈ [t] :(x i ,k) ∈ Π<br />
NP <br />
<br />
<br />
⊆ / <br />
Π <br />
P NP Π <br />
P
k<br />
<br />
<br />
<br />
G k<br />
k<br />
G k<br />
2 O(k) n O(1) <br />
k <br />
c : V → [k] <br />
k <br />
k <br />
<br />
<br />
<br />
k <br />
(G 1 ,k),...,(G t ,k)<br />
G ′ <br />
<br />
G ′ k i 1 <br />
i t G i k<br />
<br />
<br />
<br />
<br />
k k <br />
k <br />
⊆ /
P Q <br />
P Q P ppt Q<br />
f : {0, 1} ∗ −→ {0, 1} ∗ <br />
p : N → N x ∈ {0, 1} ∗ k ∈ N f(x, k) =(y, l) <br />
<br />
(x, k) ∈ P (y, l) ∈ Q <br />
l p(k)<br />
f P Q<br />
<br />
<br />
f(k)p(|x|) (x, k) f <br />
p <br />
<br />
|x| k <br />
<br />
<br />
<br />
(A, k) (B, l) A<br />
NP B ∈ NP <br />
A B B A <br />
(A, k) (B, l) <br />
A NP B ∈ NP A ppt B A <br />
B ⊆ /<br />
A <br />
NP B A
x<br />
<br />
A<br />
z ∈ A<br />
<br />
<br />
<br />
NP<br />
<br />
<br />
f(x) =y<br />
B<br />
<br />
<br />
<br />
K(y)<br />
<br />
A B A B<br />
<br />
<br />
<br />
<br />
<br />
G =(V, E) k<br />
k<br />
G k <br />
<br />
G =(V, E) k<br />
k<br />
G k
k <br />
k <br />
k k <br />
k <br />
O(k log k)<br />
<br />
O(k 2 ) <br />
O(k 3 ) <br />
<br />
<br />
<br />
<br />
<br />
k<br />
L k {1, 2, . . . , k} L ∗ k <br />
L k x ∈ L k x w 1 ···w r ∈ L ∗ k w i ···w j ; 1 <br />
i
(W, k) W = w 1 ···w n {0, 1} ∗ <br />
G =(V, E) n v 1 ,...,v n <br />
{v i ,v i+1 } 1 i
+ k <br />
n O(k 3 ) <br />
<br />
t<br />
t Π <br />
(I, k) |I| t k t (I, k) ∈ Π <br />
Π g(k) <br />
(I, k) g(k) Π<br />
(I, k) ∈ Π |I| k |x ′ |,k ′
θ c <br />
<br />
⊆ /<br />
θ c <br />
<br />
G k <br />
G k <br />
<br />
k <br />
<br />
<br />
⊆ / <br />
<br />
c θ c <br />
c c <br />
θ c <br />
θ c <br />
G k<br />
k<br />
G k H 1 ,H 2 ,...,H k <br />
1 i k H i θ c <br />
H i θ c
c = 1 θ c <br />
k G <br />
G c =<br />
2 <br />
⊆ / <br />
c 3 <br />
⊆ /<br />
<br />
<br />
<br />
G S <br />
G \ S <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
♦ k <br />
♦ k <br />
<br />
+ <br />
<br />
<br />
{θ c } <br />
<br />
θ c
{θ C } <br />
<br />
c 3 θ c <br />
k d d <br />
⊆ /<br />
<br />
<br />
<br />
θ c <br />
<br />
⊆ / <br />
<br />
k L k <br />
{1, 2, . . . , k} L ∗ k L k x ∈ L k <br />
x w 1 ···w r ∈ L ∗ k w i ···w j ; 1 i
θ c <br />
c = 5 i<br />
v j v j ; 1 j n S(v j ) (c − 2) <br />
v j D(v j )<br />
(G, k) G k <br />
<br />
θ c <br />
<br />
<br />
<br />
(w = x 1 x 2 ...x n ,k) <br />
(G, k) θ c (w, k) <br />
(w, k) <br />
(G, k) θ c <br />
l = x i , ···x j = l l w 1 l k <br />
H G {D l ,v i ,...,v j } ∪ S(v i ) <br />
θ c c <br />
D l v i H (D l ,v i ) (c − 2)<br />
〈D l ,y,v i 〉 ; y ∈ S(v i ) <br />
(D l ,v j ) v j v i G <br />
θ c l<br />
x i ...x j w <br />
l v i w
{θ C } <br />
<br />
G k <br />
θ c <br />
<br />
G k θ c <br />
D i ; 1 i k G <br />
θ c G <br />
k <br />
G <br />
l; 1 l k M l θ c G D l <br />
P l = v i ···v j L k (v i )=L k (v j )=l<br />
x i ···x j l w M l ; 1 l k P l <br />
M l w <br />
M l ; 1 l k P l <br />
1 l k<br />
M l v i ,v j <br />
L k (v i )=l = L k (v j )M l M l D l <br />
D l <br />
M l <br />
M l v i <br />
M l 2 D l <br />
M l M l<br />
v i M l v i <br />
M l G D l v i S(v i )<br />
θ c M l<br />
v i G
w L k (w) ≠ l w <br />
2 M i S(w) M i <br />
M i M i w v <br />
G <br />
M i M l v i ,v j<br />
L k (v i )=L k (v j )=l M l <br />
v i v j M l G D l v i ,v j <br />
S(v i ) ∪ S(v j ) <br />
M l D l <br />
M l <br />
v i v j
⊆<br />
/<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
(W, k) W = w 1 ···w n {0, 1} ∗ <br />
G =(V, E) n v 1 ,...,v n P<br />
G i ∈ L k x i x i<br />
v j w j = i i<br />
<br />
n u 1 ,...,u n i ∈ L k <br />
y i y i u j w j = i <br />
{u i ,u i+1 } 1 i
w = 1122312 ← <br />
y 2<br />
<br />
1 1 2 2 3 1 2<br />
u 1 s 1 u 2 s 2 u 3 s 3 u 4 s 4 u 5 s 5 u 6 s 6 u 7<br />
1 1 2 2 3 1 2<br />
v 1 r 1 v 2 r 2 v 3 r 3 v 4 r 4 v 5 r 5 v 6 r 6 v 7<br />
<br />
ppt <br />
x 2<br />
<br />
w k L G k <br />
<br />
<br />
<br />
<br />
w k L G k<br />
<br />
i ∈ L i <br />
G i w p ···w q <br />
x i v p s p u p+1 ···s q u q y i x i .<br />
Q <br />
(x i ,v p ) (v p ,s p ) (u q ,y i ) (y i ,x i ) <br />
Q <br />
Q <br />
i j w i = w j
Q <br />
(u p ,s p ) <br />
<br />
<br />
<br />
x i y i v p <br />
w p i v p <br />
w <br />
Q w p+1 ...w q <br />
<br />
w s p <br />
s p P <br />
s p s p v p P<br />
v p s p <br />
<br />
G k <br />
w k L <br />
<br />
<br />
<br />
<br />
G k w<br />
k L<br />
C 1 ,...,C k k <br />
C j x p y p 1 p k<br />
G {(x p ,y p ) | 1 p k} <br />
V(G) x p<br />
y p P Q <br />
<br />
G <br />
E C := {(x p ,y p ) | 1 p k}.<br />
C j E C <br />
k E C C 1 ,...,C k
C i (x p ,y p ) <br />
C j x p y p 1 p k<br />
{x p ,y p } Q p<br />
<br />
<br />
<br />
w = 1122312<br />
y 1<br />
y 2<br />
<br />
1 1 2 2 3 1 2<br />
u 1 s 1 u 2 s 2 u 3 s 3 u 4 s 4 u 5 s 5 u 6 s 6 u 7<br />
1 1 2 2 3 1 2<br />
v 1 r 1 v 2 r 2 v 3 r 3 v 4 r 4 v 5 r 5 v 6 r 6 v 7<br />
<br />
x 1<br />
x 2<br />
(x p ,y p )<br />
1 p k
G <br />
M S G <br />
S <br />
S M H G<br />
H M<br />
<br />
<br />
+ <br />
G <br />
3 G 4<br />
M |M| <br />
M <br />
G <br />
<br />
M <br />
G 3 <br />
|M| <br />
O ∗ (2 |M| ) <br />
<br />
<br />
3 3
⊆ / <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
k <br />
+ <br />
<br />
<br />
<br />
<br />
<br />
⊆ /
⊆ / <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
k <br />
+ <br />
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U F = {F 1 ,F 2 ,...,F n } U<br />
i, j F i ∪ F j = U <br />
C : F → U C(F i ) ∈ F i <br />
R ⊆ F ⋃ S∈R<br />
S = U C R <br />
<br />
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U F = {F 1 ,F 2 ,...,F n } U<br />
i, j F i ∪ F j = U
R ⊆ F ⋃ S∈R S = U<br />
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<br />
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<br />
U F = {F 1 ,F 2 ,...,F n } U<br />
<br />
k<br />
R ⊆ F |R| k <br />
S∈RS ⋃ = U<br />
k = 1 i F i = U <br />
k>1 ( n<br />
2) <br />
i, j F i ∪ F j = U <br />
<br />
<br />
<br />
(U, F =<br />
{F 1 ,...,F n },k) <br />
(U ′ , F ′ ,C) <br />
F ′ k F <br />
X = {x 1 ,x 2 ,...,x k } U 1 i n, 1 j k <br />
F ij = F i ∪{x j } 1 i n, 1 j k C(F ij )=x j U ′ = U∪X<br />
F ′ = {F ij | 1 i n, 1 j k} U ′ = U ∪ X <br />
i, j, p, q F ij ∪ F pq = U ′ <br />
<br />
R = {F i1 ,F i2 ,...,F it } k <br />
R ′ = {F i1<br />
∪ {x 1 },F i2 ∪<br />
{x 2 },...,F it<br />
∪ {x t }} 1 j k F ′ j = {F ij | 1 i n} t < k,<br />
F ′ j ,t < j k R′ .
R ′′ . R ′′ <br />
(U ′ , F ′ , C) d ∈ X F ′ j<br />
d R ′′ d ∈ U. d ∈ F ij ∈ R <br />
F ij R F ij ∪ {x j } ∈<br />
R ′ ⊆ R ′′ d R ′′ <br />
<br />
<br />
R ′′ R ′′ <br />
F ′ j R ′′ = {F ′′<br />
1 ,F′′ 2 ,...,F′′ k} R = {F ′′<br />
1\X, F ′′<br />
2\X, . . . , F ′′ k\X} <br />
d ∈ U F ′′<br />
i ∈ R′′ F ′′<br />
i \X ∈ R |R| k,<br />
<br />
<br />
(U, F = {F 1 ,...,F n },C)<br />
<br />
⋃<br />
S∈F S = U T F i ∈ F <br />
x ∈ F i x[i] V(T) U <br />
F i ,1 i n<br />
F i u i x[i]
F i r r <br />
u i T <br />
V(T) ={x[i] | x ∈ F i ,1 i n} ∪ {u 1 ,...,u n } ∪ {r}<br />
E(T) ={(x[i],u i ) | x ∈ F i ,1 i n} ∪ {(u i ,r) | 1 i n}<br />
<br />
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<br />
F i <br />
C(F i )<br />
x ∈ U c x <br />
f i C(F i ) c <br />
x[i],x∈ F i c x u i c fi F i <br />
u i r c r <br />
c(x[i]) = {c x | x ∈ F i ,1 i n}<br />
c(u i )={c fi | 1 i n c(r) =c r }.<br />
U <br />
c r T (T, |U| + 1, c)<br />
<br />
R = {F i1 ,F i2 ,...,F it } ⊆ F T ′<br />
T u i1 ,...,u it r <br />
(r, u i1 ),...,(r, u it ) u j T ′
u j <br />
T ′′ . <br />
T ′′ <br />
c x x ∈ U ∪ {r} T ′′ c x <br />
c x = c r T ′′ r c x ≠ c r . F p<br />
R x ∈ U <br />
C(F p )=x <br />
T ′′ u p c x C(F p ) ≠ x <br />
T ′′ T ′′ u p <br />
c x T ′′ T |U| + 1 <br />
<br />
T ′′ T |U| + 1 {u i1 ,u i2 ,...,u it } =<br />
V(T ′′ ) ∩ {u 1 ,...,u n } R = {F i1 ,F i2 ,...,F it }. <br />
R {c(u i1 ),c(u i2 ),...,c(u it )} <br />
d ∈ U T ′′ |U| + 1 <br />
T ′′ c d c d ≠ c r . T ′′ c d <br />
u i u i . F i R<br />
R d <br />
<br />
(T, C) T <br />
u 1 ,...u r T V i <br />
u i U i V i ∪ {u i } X ⊆ V(T) c(X) <br />
X <br />
c(X) ={d | ∃x ∈ X, c(x) =d}.<br />
<br />
i ≠ j<br />
(c(U i ) ∪ c(U j )) \ C ≠ φ.<br />
<br />
<br />
i, j F i ∪ F j = U
k <br />
k <br />
(T, k, c) <br />
<br />
r r T<br />
r <br />
T(i) i th r T <br />
<br />
T(i) :=T[v i ∪ ( N[v i ] \ r ) ].<br />
T(i) i th <br />
T <br />
(Q, k, c q ) <br />
(T, k, c) Q Q T <br />
( k<br />
2) <br />
V(Q) =V(T) ∪ {v[i, j] | i, j ∈ [k] i ≠ j}.<br />
X {v[i, j] | i, j ∈<br />
[k] i ≠ j} r c q (u) =c(r) <br />
u ∈ V(Q) \ V(T) u V(T) c q (u) =c(u)
Q T <br />
T(i) T(j) v[i, j] T(i) ∪ T(j)<br />
v[i, j] <br />
r v[i, j] <br />
<br />
∀{u, v},u ∈ T(i),v ∈ T(j) i, j ∈ [k] i ≠ j (u, v) ∈ E(Q) <br />
(u, v) ∈ E(T)<br />
∀i, j ∈ [k],i≠ k u ∈ T(i) ∪ T(j) (v[i, j],u) ∈ E<br />
∀u ∈ X ∀v ∈ N[r] (u, v) ∈ E<br />
∀u, v ∈ X (u, v) ∈ E<br />
Q<br />
Q <br />
<br />
T(i) T(j) i ≠ k + 1<br />
j ≠ k + 1 v[i, j] (u, v) <br />
u ∈ T(i) v ∈ T(j) v[i, j] <br />
T(i) T(j) <br />
T(l) v[l, u] u <br />
T(l) v[i, j] v[i, j] <br />
v[l, u] T(l)<br />
S<br />
v S v
(T, k, c) <br />
(Q, c q ,k) <br />
T ′ T Q R <br />
Q R<br />
R <br />
Q R \ v[i, j] T <br />
T <br />
R T ′ T <br />
<br />
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<br />
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k <br />
(k + 1) <br />
(T, c, k) k T(i) <br />
T i <br />
T(i) ={v ∈ T | c(v) =i}.<br />
T(i) i <br />
(R, c r ,k+ 1) T R <br />
(Q, c q ,k+ 1) (T, c, k) <br />
Q T ( k<br />
2) <br />
V(Q) =V(T) ∪ {v[i, j] | i, j ∈ [k] i ≠ j}.<br />
(k+1) c q (u) =k+1 u ∈<br />
V(Q) \ V(T) T(i) i Q i ∈<br />
[k + 1] u V(T) c q (u) =c(u) Q <br />
T <br />
T(i) T(j) <br />
v[i, j] T(i) ∪ T(j)
i ∈ [k + 1] ∀{u, v} ∈ T(i) (u, v) ∈ E(Q)<br />
<br />
∀{u, v},u ∈ T(i),v ∈ T(j) i, j ∈ [k] i ≠ j (u, v) ∈ E(Q) <br />
(u, v) ∈ E(T)<br />
∀i, j ∈ [k],i≠ k u ∈ T(i) ∪ T(j) (v[i, j],u) ∈ E<br />
Q<br />
Q <br />
T(i) T(i)<br />
T(j) i ≠ k + 1 j ≠ k + 1<br />
v[i, j] (u, v) u ∈ T(i)<br />
v ∈ T(j) v[i, j] <br />
T(i) T(j) T(l)<br />
v[l, l ′ ] l ′ ≠ l T(l) v[i, j]<br />
T(k + 1) v[l, l ′ ] T(l)<br />
(R, c r ,k+ 1) R <br />
Q v[i, j] ∈ T(k + 1) <br />
V[i, j] ={v[i, j] (1) ,v[i, j] (2) ,...,v[i, j] (d ij) },<br />
d ij v[i, j] V(R) \ T(k + 1) <br />
N(v[i, j]) <br />
(v[i, j] (l) ,u l ) u l l th v[i, j] <br />
T(k + 1) u V(Q) c r (u) =<br />
c q (u) u ∈ T(k + 1) c r (u) =k + 1
R<br />
R <br />
T(i) <br />
T(i) T(j) i ≠ k + 1 j ≠ k + 1 <br />
u ∈ T(i) l th<br />
u<br />
v[i, j] Q v ∈ T(j) l th<br />
v<br />
v[i, j] (u, v[i, j] l u<br />
) ∈ E (v[i, j] l u<br />
,v[i, j] l v<br />
) ∈ E (v[i, j] l v<br />
,v) ∈<br />
E u v <br />
v[i, j] l T(i) T(j) T(l)<br />
<br />
(T, c, k) <br />
(R, c r ,k + 1) <br />
T ′ T R ′ R <br />
R ′ T ′ ∪ {(u, v)} u ∈ T ′ v ∈ T(k + 1) (u, v) ∈ E<br />
v R ′ R u<br />
R ′ T(k + 1) u R ′ <br />
u ∈ T(k + 1) T(k + 1) <br />
R ′ \{u} T ′ T <br />
<br />
<br />
⊆ /<br />
<br />
⊆ /
G =(V, E) <br />
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(T, k, c) <br />
T <br />
c : V(T) → [k]
T p T q l p ∈ V(T p ) <br />
T p f q ∈ V(T q ) T q T p ⊙ T q <br />
<br />
V(T p ⊙ T q )=V(T p ) ⊎ V(T q ) ∪ {v p ,v q } {v p ,v q } <br />
E(T p ⊙ T q )=E(T p ) ⊎ E(T q ) ∪ {(l p ,v p ), (v p ,v q ), (v q ,f q )}<br />
T p ⊙ T q <br />
<br />
<br />
⊆ /<br />
(T 1 ,c 1 ,k), (T 2 ,c 2 ,k),...(T t ,c t ,k) <br />
T <br />
T = T 1 ⊙ T 2 ⊙ ···⊙ T t .<br />
N ⊙ <br />
T N T i <br />
T T(T i )<br />
N T i <br />
T i T i <br />
c T u ∈ T i c(u) =c i (u) <br />
u T i c(u) =c(v) v T i u<br />
(T,k) <br />
∃i, i ∈ [t] T i <br />
T <br />
T i T
T R <br />
T V(R) T(T i ) T(T j )<br />
i ≠ j <br />
V(R) T i T j <br />
<br />
T i T j <br />
V(R) T(T i ) i 1 i t R <br />
T i <br />
⊆ /<br />
<br />
⊆ /<br />
<br />
<br />
<br />
<br />
<br />
<br />
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n O(k 2 )<br />
<br />
<br />
<br />
<br />
G =(V, E) k ∈ N c : V → [k] r ∈ V<br />
k<br />
G T k r c <br />
T
T =(V, E) (T, k, c, u) <br />
T T u<br />
O(k 2 ) T u<br />
T T k u <br />
T u <br />
k u<br />
T u O(k 2 ) k<br />
u T 2k <br />
k u <br />
2k u <br />
k u T u <br />
2k k <br />
2k T u<br />
2k+2k·k+2k = O(k 2 ) n u T <br />
n <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
O(k 2 )
(T, k, c) T {v 1 ,v 2 ,...,v l }<br />
T 1 T l T 1 ,T 2 ,...,T l T <br />
v 1 i l v v i T i <br />
T ′ c ′ T <br />
T k + 1 v (T ′ ,k+ 1, c ′ ,v) <br />
T ′ <br />
(T, k, c) (T ′ ,k+ 1, c ′ ,v)<br />
<br />
<br />
⊆ /<br />
<br />
<br />
<br />
(T 1 ,v 1 ,c 1 ,k), (T 2 ,v 2 ,c 2 ,k),...(T r ,v r ,c r ,k) T i =<br />
(V i ,E i ) T =(V, E) <br />
V = {u} ∪ {u 1 ,u 2 ,...,u r }∪ i∈[r] V i<br />
(u, u i ) ∈ E i ∈ [r] (u i ,v i ) ∈ E i ∈ [r]<br />
E i ⊂ E i ∈ [t]<br />
c : V → [k + 2] v ∈<br />
V i c(v) =c i (v) c(u i )=k + 1 u i c(u) =k + 2<br />
(T,k + 2, c, v) T i <br />
v i k + 2
v v i (v i ,u i ) (u i ,u) <br />
T T ′ u <br />
T ′ T i T ′ <br />
T i T j T ′ u i u j <br />
T ′ <br />
<br />
<br />
<br />
<br />
<br />
<br />
G =(V, E) c : V → [k] <br />
U ⊆ V |U| = s = O(1)<br />
k<br />
G T k U ⊆ V(T) c<br />
T <br />
<br />
⊆ /
(T, v, c, k) T =(V, E) <br />
s ′ = s − 1 T ′ =(V ′ ,E ′ ) T ′ =(V ′ ,E ′ )<br />
V ′ = V ∪ {u 1 ,u 2 ,...,u ′ s} E ′ = E ∪ {(v, u 1 ),...,(v, u ′ s)}<br />
c ′ : V ′ → [k + s ′ ] <br />
c ′ (v) =c(v) ∀v ∈ V<br />
c ′ (u j )=k + j ∀j ∈ [s ′ ]<br />
(T ′ ,c ′ , {v, u 1 ,...,u ′ s},k+ s ′ ) <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
(G, k, c) <br />
C(i) C i <br />
C(i) ={v ∈ G | c(v) =i}. C(i) i<br />
(H, k) <br />
<br />
<br />
S S ⊆ V (u, v) u, v ∈ S u ≠<br />
v
(S, T) S, T ⊆ V S ∩ T = ∅ <br />
(u, v) u ∈ S v ∈ T<br />
(S, T) S, T ⊆ V S ∩ T = ∅ |S| = |T| <br />
S T S = {s 1 ,s 2 ,...,s r }<br />
T = {t 1 ,t 2 ,...,t r } (s i ,t i ) i 1 i r<br />
u S u S ⊆ V (u, v)<br />
v ∈ S<br />
H G H<br />
G H<br />
H = G<br />
C(i) i ∈ [k]<br />
C(i) (k + 1) v i (1),v i (2),...,v i (k + 1)<br />
H(i) (C(i),H(i))<br />
(i, j) i ≠ j (H(i),H(j)) <br />
H i H j <br />
j th H i v i (j) <br />
(v i (p),v j (q)) p ≠ q <br />
D(i, j) D = ∪ P D(i, j) P = ( )<br />
[k]<br />
2 <br />
D <br />
u ∈ D(i, j) u C(l) l l ≠ i l ≠<br />
j<br />
H <br />
<br />
u w <br />
<br />
<br />
u ∈ C(i) w ∈ C(i) u w <br />
C(i)
u ∈ C(i) w ∈ C(j) j ≠ i <br />
<br />
<br />
u ∈ H(i) w ∈ H(i) <br />
x ∈ C(i) (u, x) (w, x) <br />
<br />
u ∈ H(i) v ∈ H(j) i ≠ j u = v i (p) w = v j (q) p = q <br />
u v <br />
u w <br />
<br />
u ∈ H(i) w ∈ C(i) (u, w) <br />
<br />
<br />
u ∈ H(i) w ∈ C(j) j ≠ i u = v i (p) (u, v j (p))
C H D<br />
C G D <br />
D <br />
H D <br />
D D <br />
D <br />
<br />
(v j (p),w) <br />
<br />
u ∈ D w ∈ D u w D<br />
<br />
<br />
<br />
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u ∈ D(i, j) w ∈ C(l) l ≠ i l ≠ j <br />
u w u C(l) <br />
u ∈ D(i, j) w ∈ C(i) w ∈ C(j) u <br />
H(i) H(j) w <br />
<br />
u ∈ D w ∈ H(j) j u w <br />
w w D<br />
D <br />
H <br />
G H <br />
H(i) i ∈ [k] <br />
H(i) C(i) <br />
C(i) D <br />
<br />
k G
C(i) S <br />
k <br />
|S ∩ C(i)| 1.<br />
|S| k |S ∩ C(i)| = 1 S <br />
S |S ∩<br />
C(i)| 1 <br />
|S ∩ C(i)| = 0<br />
u/∈ C(i) H(i) <br />
u ∈ H(i) H(i) u <br />
<br />
u ∈ D u H(i) <br />
H(i)<br />
u ∈ H(j) j ≠ i u H(i) <br />
H(i)<br />
H(i)<br />
C(i) <br />
k H(i) S<br />
k
C(i) S <br />
k <br />
|S ∩ C(i)| 1.<br />
|S| k |S ∩ C(i)| = 1 S <br />
S |S ∩<br />
C(i)| 1 <br />
|S ∩ C(i)| = 0<br />
u/∈ C(i) H(i) <br />
u ∈ H(i) H(i) u <br />
<br />
u ∈ D u H(i) <br />
H(i)<br />
u ∈ H(j) j ≠ i u H(i) <br />
H(i)<br />
H(i)<br />
C(i) <br />
k H(i) S<br />
k
F <br />
F<br />
F <br />
F <br />
p p<br />
<br />
F <br />
♦ F <br />
♦ F = {O 1 ,...,O p } <br />
<br />
♦ F = {K 5 ,K 3,3 } <br />
<br />
<br />
F F
⋆ F O(log 3/2 OPT) <br />
F <br />
⋆ t <br />
K 1,t F <br />
F <br />
⋆ F θ c <br />
c F <br />
<br />
⋆ F <br />
F <br />
<br />
F <br />
F <br />
⋆ <br />
F 2 O(k log k) n O(1) <br />
(2 kO(1) n O(1) ) <br />
<br />
<br />
<br />
P P <br />
<br />
<br />
P <br />
<br />
G <br />
H O( √ |V(G)|) <br />
F F <br />
O( √ n)
F<br />
<br />
♦ F <br />
F <br />
♦ F F <br />
n ( 1 2 −ε) ε >0<br />
F <br />
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F <br />
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F<br />
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F<br />
F = { θ c } <br />
F <br />
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θ c <br />
<br />
θ c <br />
<br />
k <br />
k θ c <br />
k <br />
k <br />
k<br />
<br />
♦ H M(H) <br />
<br />
♦ H M(H)
n <br />
<br />
<br />
<br />
<br />
<br />
⋆ <br />
O(k 2 ) n <br />
O(k 2 ) <br />
⋆ <br />
⊆ /
n <br />
<br />
<br />
<br />
<br />
<br />
⋆ <br />
O(k 2 ) n <br />
O(k 2 ) <br />
⋆ <br />
⊆ /
n <br />
<br />
<br />
<br />
<br />
<br />
⋆ <br />
O(k 2 ) n <br />
O(k 2 ) <br />
⋆ <br />
⊆ /
+ <br />
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k O(n 2 ) <br />
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W[1] <br />
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H<br />
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+ <br />
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<br />
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+
+ <br />
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+ <br />
+ <br />
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H
O( √ |V||E|)
O( √ |V||E|)
O( √ |V||E|)