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arXiv · 1506.04380

Structure of Graphs with Locally Restricted Crossings

Vida Dujmović·David Eppstein·David R. Wood
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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 $O(\sqrt{(g+1)(k+1)n})$ and layered treewidth $O((g+1)k)$, and that these bounds are tight up to a constant factor. As a special case, the $k$-planar graphs with $n$ vertices have treewidth $O(\sqrt{(k+1)n})$ and layered treewidth $O(k+1)$, which are tight bounds that improve a previously known $O((k+1)^{3/4}n^{1/2})$ treewidth 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.

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Vida Dujmović, David Eppstein, David R. Wood. 2015-06-14. Structure of Graphs with Locally Restricted Crossings. https://arxiv.org/abs/1506.04380

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