Algorithm Design

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Chapter 10 Extending the Limits of Tractability
Gt l
No edge (u, v)
Figure 10.6 Separations of the tree T translate to separations of the graph G.
Proof. We refer to Figure 10.6 for a general view of what the separation looks
like. We first prove that the subgraphs GT~--Vt do not share any nodes. Indeed,
any such node v would need to belong to both GT~--Vt and GTj--Vt for some
i #j, and so such a node u belongs to some piece V x with x ~ T i, and to some
piece Vy with y ~ Tj. Since t lies on the x-y path in T, it follows from the
Coherence Property that u lies in V t and hence belongs to neither.Gw~- Vt nor
GT;-- Vt.
Next we must show that there is no edge e = (u, v) in G with one end u
in subgraph GT~--Vt and the other end v in GT;--Vt for some j # i. If there
were such an edge, then by the Edge Coverage Property, there would need to
be some piece V x containing both u and v. The node x cannot be in both the
subgraphs Ti and Tj. Suppose by symmetry x ~ T~. Node u is in the subgraph
GT~, so u must be in a set Vy for some y in T~. Then the node u belongs to both
V x and Vy, and since t lies on the x-y path in T, it follows that u also belongs
to V t, and so it does not lie in GT~--Vt as required. []
Proving the edge separation property is analogous. If we delete an edge
(x, y) from T, then T falls apart into two components: X, containing x, and Y,
No edge (u, u)
Gx - v.~ n ~, G~,- v.,: n v),
10.4 Tree Decompositions of Graphs
Figure 10.7 Deleting an edge of the Ixee T translates to separation of the graph G.
containing y. Let’s establish the corresponding way in which G is separated
by this operation.
(10.14) Let X and Y be the two components ofT after the deletionof the edge
(x, y). Then deleting the set V.,: ¢~ V v from V disconnects G into the two subgraphs
Gx - (V x N Vy) and Gy- (g x n V~,i More precisely, these two subgraphs do not
share any nodes, and there is no edge with one end in each of them.
Proof. We refer to Figure 10.7 for a general view of what the separation looks
proves like. The that proof the two of this subgraphs property Gx- is (V analogous x ~ Vy) and to the Gy proof -- (gx of n vp (!0.13). do not One share first
any nodes, by showing that a node ~ that belongs to both G x and Gz must
belong to both V x and to Vy, and hence it does not lie in either Gy-(V x ~ Vy)
or Gx- (Vx n vp.
Now we must show that there is no edge e = (u, u) in G with one end u
in G x- (V x ~ Vy) and the other end ~ in G~,- (v x n vy). If there were such an
edge, then by the Edge Coverage Property, there would need to be some piece
Vz containing both a and v. Suppose by symmetry that .z ~ X. Node v also
belongs to some piece Vw for w ~ Y. Since x and y lie on the w-z path in T, it
follows that V belongs to V x and Vy. Hence u ~ V x ~ Vy, and so it does not lie
in Gy- (V x n Vy) as required. []
So tree decompositions are useful in that the separation properties of T
carry over to G. At this point, one might flfink that the key question is: Which
graphs have tree decompositions? But this is not the point, for if we think about
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