Shortest Path Embeddings of Graphs on Surfaces

摘要

AbstractThe classical theorem of Fáry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fáry’s theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property ashortest path embedding. The main question addressed in this paper is whether given a closed surfaceS, there exists a Riemannian metric for which every topologically embeddable graph admits a shortest path embedding. This question is also motivated by various problems regarding crossing numbers on surfaces. We observe that the round metrics on the sphere and the projective plane have this property. We provide flat metrics on the torus and the Klein bottle which also have this property. Then we show that for the unit square flat metric on the Klein bottle there exists a graph without shortest path embeddings. We show, moreover, that for largeg, there exist graphsGembeddable into the orientable surface of genusg, such that with large probability a random hyperbolic metric does not admit a shortest path embedding ofG, where the probability measure is proportional to the Weil–Petersson volume on moduli space. Finally, we construct a hyperbolic metric on every orientable surfaceSof genusg, such that every graph embeddable intoScan be embedded so that every edge is a concatenation of at mostO(g) shortest paths.]]>