On Hardness of the Joint Crossing Number

The Joint Crossing Number problem asks for a simultaneous embedding of two disjoint graphs into one surface such that the number of edge crossings (between the two graphs) is minimized. It was introduced by Negami in 2001 in connection with diagonal flips in triangulations of surfaces, and subsequently investigated in a general form for small-genus surfaces. We prove that all of the commonly considered variants of this problem are NP-hard already in the orientable surface of genus 6, by a reduction from a special variant of the anchored crossing number problem of Cabello and Mohar

Disjoint Essential Cycles

AbstractGraphs that have two disjoint noncontractible cycles in every possible embedding in surfaces are characterized. Similar characterization is given for the class of graphs whose orientable embeddings (embeddings in surfaces different from the projective plane, respectively) always have two disjoint noncontractible cycles. For graphs which admit embeddings in closed surfaces without having two disjoint noncontractible cycles, such embeddings are structurally characterized

Embeddings of Harary Graphs in Orientable Surfaces

The purpose of this thesis is to study embeddings of Harary graphs in orientable surfaces. In particular, our goal is to provide a complete description of one method of constructing a maximal embedding in an orientable surface for any Harary graph. Rotation systems, which describe the ordering of edges around the vertices of a graph, can be used to represent graph embeddings in orientable surfaces. Together with the Boundary Walk Algorithm, this representation provides a method of constructing a corresponding graph embedding. By switching adjacent edges in a rotation system, we can control the genus of the constructed embedding surface. We will explore how certain series of adjacent edge switches may be used to take standard rotation systems (which will be defined) to rotation systems corresponding to maximal embeddings of Harary graphs