2024/03/18 by Bulavka, Denys, de Verdière, Éric Colin, Colin de Verdière, Éric +1
#05C10 #57M15 #68Q25 #68W05 #Computational Complexity (cs.CC) #Computational Geometry (cs.CG) #F.2.2 #FOS: Computer and information sciences #FOS: Mathematics #G.2.2 #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2403.11749
We initiate the study of computing shortest non-separating simple closed curves with some given topological properties on non-orientable surfaces. While, for orientable surfaces, any two non-separating simple closed curves are related by a self-homeomorphism of the surface, and computing shortest such curves has been vastly studied, for non-orientable ones the classification of non-separating simple closed curves up to ambient homeomorphism is subtler, depending on whether the curve is one-sided or two-sided, and whether it is orienting or not (whether it cuts the surface into an orientable one). We prove that computing a shortest orienting (weakly) simple closed curve on a non-orientable combinatorial surface is NP-hard but fixed-parameter tractable in the genus of the surface. In contrast, we can compute a shortest non-separating non-orienting (weakly) simple closed curve with given sidedness in gO(1).nlog n time, where g is the genus and n the size of the surface. For these algorithms, we develop tools that can be of independent interest, to compute a variation on canonical systems of loops for non-orientable surfaces based on the computation of an orienting curve, and some covering spaces that are essentially quotients of homology covers.