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Erschienen in:
2005 | OriginalPaper | Buchkapitel
Finding Shortest Non-separating and Non-contractible Cycles for Topologically Embedded Graphs
verfasst von : Sergio Cabello, Bojan Mohar
Erschienen in: Algorithms – ESA 2005
Verlag: Springer Berlin Heidelberg
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We present an algorithm for finding shortest surface non-separating cycles in graphs with given edge-lengths that are embedded on surfaces. The time complexity is
O
(
g
3/2
V
3/2
log
V
+
g
5/2
V
1/2
), where
V
is the number of vertices in the graph and
g
is the genus of the surface. If
g
=
o
(
V
1/3 −
ε
), this represents a considerable improvement over previous results by Thomassen, and Erickson and Har-Peled. We also give algorithms to find a shortest non-contractible cycle in
O
(
g
$^{O({\it g})}$
V
3/2
) time, improving previous results for fixed genus.
This result can be applied for computing the (non-separating) face-width of embedded graphs. Using similar ideas we provide the first near-linear running time algorithm for computing the face-width of a graph embedded on the projective plane, and an algorithm to find the face-width of embedded toroidal graphs in
O
(
V
5/4
log
V
) time.