Genus, treewidth, and local crossing number

Abstract

We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an n-vertex graph embedded on a surface of genus g with at most k crossings per edge has treewidth (Formula presented.) and layered treewidth (Formula presented.), and that these bounds are tight up to a constant factor. A a special case, the k-planar graphs with n vertices have treewidth (Formula presented.) and layered treewidth O(k + 1), which are tight bounds that improve a previously known (Formula presented.) treewidth bound. Additionally, we show that for g < m, every m-edge graph can be embedded on a surface of genus g with (Formula presented.) crossings per edge, which is tight to a polylogarithmic factor.