The 2 and 3 representative projective planar embeddings

AbstractA graph embedded on a surface is n-representative if every nontrivial closed curve in the surface which does not intersect edges of the embedding must contain at least n vertices of the graph. The property of being n-representative on a surface is closed upward under minor inclusion; hence, by the results of N. Robertson and P. D. Seymour (Graph minors. VIII. A Kuratowski theorem for general surfaces, submitted for publication), the set of minor minimal n-representative embeddings on a surface is finite up to isomorphism. The property of being minor minimal n-representative is invariant under Y-Δ operations. The set of minor minimal 2 and 3 representative embeddings on the projective plane are found. These embeddings are used to produce the topologically minimal 2 and 3 representative projective embeddings

Shortest path embeddings of graphs on surfaces

The classical theorem of F\'{a}ry 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 F\'{a}ry'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 a shortest path embedding. The main question addressed in this paper is whether given a closed surface S, 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 large g, there exist graphs G embeddable into the orientable surface of genus g, such that with large probability a random hyperbolic metric does not admit a shortest path embedding of G, where the probability measure is proportional to the Weil-Petersson volume on moduli space. Finally, we construct a hyperbolic metric on every orientable surface S of genus g, such that every graph embeddable into S can be embedded so that every edge is a concatenation of at most O(g) shortest paths.Comment: 22 pages, 11 figures: Version 3 is updated after comments of reviewer

Projective plane embeddings of polyhedral pinched maps

We give various conditions on pinched-torus polyhedral maps which are necessary for their graphs to be embeddable in the projective plane. Our other main result is that even if the graph of a polyhedral map in the pinched torus is embeddable in a projective plane, the map induced by the embedding cannot be polyhedral, but must have all faces bounded by cycles. Finally, we give a class of examples of graphs which have polyhedral embeddings on the pinched torus and also on orientable surfaces of arbitrary high genus