Structure of Graphs with Locally Restricted Crossings
Vida Dujmović, David Eppstein, David R. Wood · SIAM Journal on Discrete Mathematics · 2017
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