Sparse Outerstring Graphs Have Logarithmic Treewidth

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 $O(α\log n)$, where $α$ denotes the arboricity of the graph, with an almost matching lower bound of $Ω(α\log (n/α))$. 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.

Figures (7)

• Figure 1: (a) A $\sqrt{n}\times \sqrt{n}$ grid is a unit disk graph of treewidth $\Theta(\sqrt n)$. (b) A sparse axis-parallel segment graph of treewidth $\Theta(\sqrt n)$. It does not contain $K_{2,2}$ as a subgraph. The horizontal segments form $\sqrt n$ rows, and each row consists of $\Theta(\sqrt n)$ horizontal segments.
• Figure 2: We can always assume the general position assumption.
• Figure 3: (a) The red region depicts $U_2$. Jordan curve $R_2$ consists of the parts of four strings. (b) All black strings are contained in both $\Gamma_i$ and $\Gamma_{i-4\alpha}$, and the red string is contained in $\Gamma_i$ only.
• Figure 4: (a) Illustration of an $(A,B)$-linkage. (b) Partition of $X$ into $X_1,X_2,X_3$ and $X_4$. (c) The case that $C_\pi \subseteq C_{\pi'}$. An internal vertex of $\pi'$ (corresponding to the pink dotted curve) intersects the geometric representation of $\pi$.
• Figure 5: (a) Illustration of $\tilde{X}_1, \tilde{X}_2, \tilde{X}_3$ and $\tilde{X}_4$. (b) Illustration of length-two paths of $\mathcal{P}_v$ and $\mathcal{P}_h$. Two geometric representations of two pairs of paths of $\mathcal{P}_h$ and $\mathcal{P}_v$ intersect.

Theorems & Definitions (35)

• Lemma 1
• proof
• Definition 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4: Folklore
• proof
• Lemma 5