O(1)-Distortion Planar Emulators for String Graphs

TL;DR

The paper proves that every unweighted string graph $G$ admits an $O(1)$-distortion planar emulator $H$, with $V(G)\subseteq V(H)$ and for all pairs $u,v$, $\delta_G(u,v) \le \delta_H(u,v) \le c\cdot\delta_G(u,v)$ for a universal constant $c$. The construction leverages region contact graphs on a planar base, builds a hierarchical clustering with bounded diameter and scattering properties, and contracts clusters to obtain planarity while preserving distances up to a constant factor; Steiner-point removal can further yield an emulator on the original vertex set with polynomial-time computability from the representation. The approach adapts the planar shortcut-partition framework to the more complex interactions of string graphs, via SelectPaths to handle outer regions and ClusterOuter to expand and partition while maintaining connectivity and distortion guarantees. A parallel independent result shows quasi-isometry between string graphs and planar graphs, reinforcing the central claim that string-graph metrics can be embedded into planar-like metrics with constant distortion, with implications for distance sketches and related geometric algorithms on string graphs.

Abstract

We show that every unweighted string graph $G$ has an $O(1)$-distortion planar emulator: that is, there exists an (edge-weighted) planar graph $H$ with $V(H) = V(G)$, such that every pair of vertices $(u,v)$ satisfies $δ_G(u,v) \le δ_H(u,v) \le O(1) \cdot δ_G(u,v).$

Figures (6)

• Figure 1: (Left) A string graph $G$; (Middle) A partition of $\mathbb{R}^2$ into clusters; (Right) A planar emulator for $G$ obtained by contracting all clusters and adding stars.
• Figure 2: Left: A set of regions $\mathcal{R}$ on graph $G$, and the corresponding contact graph $G_\mathcal{R}$. Right: The regions $\mathcal{R}'$ produced by $\textsc{Shatter}({\mathcal{R} - \partial\! S})$, and the resulting contact graph $G_{\mathcal{R}'}$. The subregions of region $R$ are $R_1, R_2, R_3$.
• Figure 3: A depiction of the proof of \ref{['lem:hierarchy-scattering']}. There is a plane graph $G$, and a set of clusters $\mathcal{C}$ (from \ref{['lem:gridtree']}) drawn in green. The orange region $R'$ is an outer region in $G$ and is contained entirely in $V(\mathcal{C})$. The red region $R$ is not an outer region and not contained entirely in $V(\mathcal{C})$; observe that after shattering $\mathcal{R}^* \gets \textsc{Shatter}({R_2 - \partial\! (V(\mathcal{C}))})$, every subregion of $R$ in $\mathcal{R}^*$is an outer region in the subgraph induced by the vertices not in $V(\mathcal{C})$.
• Figure 4: Left: Subgraph $H$ outer-bounded by $W$ in $H_0$, with 2 critical vertices (as described in \ref{['lem:blt']}(1). Center: The path $\pi$ separates $W$ from the outer vertices of $H_0$ in $H$ (as described in \ref{['lem:blt']}(2)). Right: A connected component $H'$ of $H \setminus W'$ is outer-bounded by $W'$ in $H_0$ (as described in \ref{['lem:blt']}(3))
• Figure 5: Illustration for ClusterOuter$(H,\mathcal{R})$. Top left: After Step 2. The dark gray regions are within $S_a$ and $S$, and the light gray regions are within $S_a^+$ and $S^+$. The colorful regions are net points. Observe that $V(S_a^+) \cap V(S^+) \neq \varnothing$. Top middle: After Step 3(a). Top right: After Step 3(b). Bottom left: After Step 4(a) processes $S_a^+$. Notice that $\textsc{Assign}(v \rightsquigarrow C_{\textrm{pink}}, V(B_{\textrm{pink}}^{\rm big}))$ actually assign $v$ to the blue cluster. Bottom right: After Step 4(b) processes $S^+$. Observe that $\textsc{Assign}(v \rightsquigarrow C_{\textrm{green}}, V(B_{\textrm{green}}^{\rm big}))$ assign $v$ to the blue cluster.

Theorems & Definitions (46)

• Theorem 1.1
• Lemma 2.1
• proof : of \ref{['thm:main']}
• Lemma 3.1
• proof
• Lemma 3.4
• proof
• Lemma 4.1
• proof
• Claim 4.3