Schnyder woods for higher genus triangulated surfaces with ...

9 Schnyder woods for higher genus triangulated surfaces
S
S out S in
e m
w ′ e m
v w
u ′
u
v w
w ′′
e s
u
w
e s
e m
Figure 7: An example of execution of our traversal algorithm for a triangulated surface of genus 1. As in the planar case the traversal starts
from the root face (the one incident to the green root edge). As far as only conquer operations on free vertices are performed, the area already
explored (white triangles) remains planar and homeomorphic to a disk (second picture). When there remain no free vertices to conquer (third
picture), it is possible to find split and merge edges (incident to black circles). After performing split and merge operations, (here on the edges
(u, w) and (w, v)), new free vertices can be found (here u ′ , w ′ , and w ′′ ) in order to continue the traversal, until the entire graph is visited.
neighboring vertices lying in the opposite side of the sector
(e 1 ,e 2 ), respect to the edge e 2 (recall that e 1 and e 2 could
coincide, when w has multiplicity 2 being incident to only
one special edge).
Consider a g Schnyder wood computed by our algorithm,
and let C be the cut-graph of the Schnyder wood. In the
statement of the following lemma, let C be the part of
C already conquered at some step of the algorithm (the
sub-graph contained in S in ). Each connected area of S\C
is called a face of C (though C might not be a cellular
embedding, i.e., some of these faces might not be topological
disks).
Lemma 11 At each step (strictly before the end) of the algorithm
computing a g Schnyder wood, let C be the part of
the cut graph already discovered. Then there is exactly one
cycle of the boundary of S in inside each facial walk of C.
Moreover S in \C is a planar surface.
Proof : The property is true initially: S in is the root face,
which is planar, and C is reduced to v n−1 , which has the
total surface S as its unique face. The property remains also
true at each vertex conquest, since each conquest simply
increases the area of S in within a facial walk W of C, but
does not close the boundary of S in within W unless we are
at the end of the algorithm. If the boundary is closed, the
area of the surface S within W will be a face in C, hence
that face must be the unique face of C, i.e., the algorithm
is at its end. Otherwise the interior of that face would be a
contractible area, so the final cut-graph would have a face
with a contractible boundary, which is impossible. For each
split, we cut a cycle C of the boundary of S in into two
parts C 1 and C 2 , but the special edge addition also cuts the
corresponding facial walk W of C into two facial walks
W 1 and W 2 that contain respectively C 1 and C 2 . The genus
of S in does not change during a split, so S in \ C remains
planar. For each merge, we merge two cycles C 1 and C 2 of
the boundary of S in into a unique cycle by means of a merge
edge that establishes a bridge between the two components.
The corresponding facial walks W 1 and W 2 of C are also
merged into a unique facial walk W by means of the special
edge, and W contains C in its interior, as illustrated in
Figure 10. It is also easily observed on that figure that the
two boundary cycles previously enclosed inside W 1 and W 2
have been merged into a unique boundary cycle inside W .
Hence the surface S in \ C remains planar.
Proposition 12 The algorithm COMPUTESCHNYDERANYGENUS(S)
computes a g-Schnyder wood.
Proof : As previously seen, the algorithm terminates and
the orientations and colorations of edges satisfy the local
properties of a g-Schnyder wood. It remains to show the cutgraph
property. First let us show that T 2 is a tree spanning all
vertices except v 0 and v 1 . By the local conditions, all vertices
except those of the root face have one outgoing edge of color
Preprint MAT. 18/08, communicated on September 1 st , 2008 to the Department of Mathematics, Pontifícia Universidade Católica — Rio de Janeiro, Brazil.