Sparse Outerstring Graphs Have Logarithmic Treewidth

TL;DR

This paper proves that outerstring graphs have logarithmic treewidth relative to their size and arboricity, enabling efficient algorithms for many NP-complete problems on these graphs.

Contribution

It establishes tight bounds on the treewidth of outerstring graphs based on arboricity and applies these results to develop faster algorithms for NP-complete problems.

Findings

Outerstring graphs have treewidth O(α log n).

Polynomial-time algorithms for NP-complete problems on sparse outerstring graphs.

Subexponential algorithms for NP-complete problems on general outerstring graphs.

Abstract

An outerstring graph is the intersection graph of curves lying inside a disk with one endpoint on the boundary of the disk. We show that an outerstring graph with $n$ vertices has treewidth $O(\alpha\log n)$, where $\alpha$ denotes the arboricity of the graph, with an almost matching lower bound of $\Omega(\alpha \log (n/\alpha))$. As a corollary, we show that a $t$-biclique-free outerstring graph has treewidth $O(t(\log t)\log n)$. This leads to polynomial-time algorithms for most of the central NP-complete problems such as \textsc{Independent Set}, \textsc{Vertex Cover}, \textsc{Dominating Set}, \textsc{Feedback Vertex Set}, \textsc{Coloring} for sparse outerstring graphs. Also, we can obtain subexponential-time (exact, parameterized, and approximation) algorithms for various NP-complete problems such as \textsc{Vertex Cover}, \textsc{Feedback Vertex Set} and \textsc{Cycle Packing} for (not necessarily sparse) outerstring graphs.