Structure of Graphs with Locally Restricted Crossings

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 $O(\sqrt{(g+1)(k+1)n})$ and layered treewidth $O((g+1)k)$ and that these bounds are tight up to a constant factor. In the special case of $g=0$, so-called $k$-planar graphs, the treewidth bound is $O(\sqrt{(k+1)n})$, which is tight and improves upon a known $O((k+1)^{3/4}n^{1/2})$ bound. Analogous results are proved for map graphs defined with respect to any surface. Finally, we show that for $g<m$, every $m$-edge graph can be embedded on a surface of genus $g$ with $O((m/(g+1))\log^2 g)$ crossings per edge, which is tight to a polylogarithmic factor.