Finding Shortest Non-separating and Non-contractible Cycles for Topologically Embedded Graphs

Abstract

We present an algorithm for finding shortest surface non-separating cycles in graphs embedded on surfaces in O(g 3/2 V 3/2 log V + g 5/2 V 1/2 ) time, 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 HarPeled. We also give algorithms to find a shortest non-contractible cycle in O(g O(g) V 3/2 ) time, which improves 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.

The crossing number of a projective graph is quadratic in the face–width

“…For that one can use an O(n √ n)-time algorithm by Cabello and Mohar [4]. As pointed to us by S. Cabello [private communication], the same goal can be achieved in O(n log n) time using a suitable preprocessing and then algorithm of Klein [9] (for planar distances).…”

Dynamic Programming for Graphs on Surfaces

“…Similarly to [6], we use the notation GSN for the graph obtained by cutting G along the noose N and gluing a disk on the obtained boundaries.…”

Schnyder Woods for Higher Genus Triangulated Surfaces, with Applications to Encoding

“…Actually, given a surface S of genus g and size n, one can compute a cut-graph or a collection of 2g non-trivial cycles, whose removal makes S a topological disk (possibly with boundaries). There is a number of recent works [8,15,16,24,25,35] dealing with interesting algorithmic challenges concerning the efficient computing of cut-graphs, optimal (canonical) polygonal schemas and shortest non-trivial cycles. For example some works make it possible to compute polygonal schemas in optimal O(gn) time for a triangulated orientable manifold [25,35].…”

“…From the combinatorial point of view this would imply to deal with boundaries of arbitrary size (arising from the planarizing procedure), as non-trivial cycles can be of size Ω( √ n), and cut-graphs have size O(gn). Moreover, from the algorithmic complexity point of view, the most efficient procedures for computing small non-trivial cycles [8,24] require more than linear time, the best known bound being currently of O(n log n) time.…”