$K_{2,3}$-induced minor-free graphs admit quasi-isometry with additive distortion to graphs of tree-width at most two

TL;DR

The paper proves that $K_{2,3}$-induced minor-free graphs admit a quasi-isometry with additive distortion to graphs of tree-width at most $2$, via a constructive embedding that preserves distances up to a fixed constant. By leveraging layering partitions and the bounded strong isometric path complexity of this graph class, the authors design an $O(nm)$-time algorithm that produces a graph $H$ with $V(H)=V(G)$, $ ext{tw}(H)\\le 2$, and $|d_G(u,v)-d_H(u,v)|\le c$ for all pairs $u,v$. For universally signable graphs, the algorithm runs in linear time, enabling a truly sub-quadratic additive-constant approximation for the diameter, contrasted with SETH-based hardness for related graph classes. The results unify structural graph theory with coarse metric techniques and open avenues for extending to broader $K_{2,r}$-induced minor-free families and other even-hole/pyramid-related classes.

Abstract

A graph $H$ is an \emph{induced minor} of a graph $G$ if $H$ can be obtained from $G$ by a sequence of edge contractions and vertex deletions. Otherwise, $G$ is \emph{$H$-induced minor-free}. In this paper, we provide a different proof of the fact that $K_{2,3}$-induced minor-free graphs admit a quasi-isometry with additive distortion to graphs with tree-width at most two. Our proof yields a $O(nm)$-time algorithm which takes as input a $K_{2,3}$-induced minor-free graph with $n$ vertices and $m$ edges, and outputs a tree-width two graph $H$ with the desired additive distortion. For \emph{universally signable} graphs, a subclass of $K_{2,3}$-induced minor-free graphs, the time complexity of our algorithm is linear. As a consequence, we obtain a truly sub-quadratic time additive constant factor approximation algorithm to compute the \emph{diameter} of a universally signable graph. In contrast, assuming the \emph{Strong Exponential Time Hypothesis} (\textsc{SETH}), the diameter of split graphs (a very restricted class of universally signable graphs), cannot be computed in truly sub-quadratic time [Borassi et al. (ENTCS, 2016)].

Figures (3)

• Figure 1: Illustration of the proof of \ref{['lem:sipco-distort']}. Here $z=3$. The dashed curve from $u$ to $v$ indicates a $(u,v)$-path in $Y$. The solid black piecewise linear curve between $(u,v)$ that doesn't pass through $r$, indicates the $(u,v)$-isometric path $P$ in $X$. For $p\in \{a_1=u, b_2, b_3 =v \}$, the strictly monotone solid black curves between $r,p$ indicate a $(r,p)$-isometric path in $X$.
• Figure 2: The shaded regions indicate the clusters w.r.t $r$. The graph $H$ is the output of \ref{['algo:embed']} with $G$ and $r$ as input.
• Figure 3: Illustration of the proof of \ref{['lem:cycle']}. The dashed lines indicate edges in $H$. The solid lines indicate edges in $G$. The curved lines indicate induced paths in $G$. The vertices in the shaded region indicate cliques in $G$. There are no edges in $G$ between a vertex in a rectangular region and a vertex in a elliptical region.

Theorems & Definitions (65)

• Theorem 1: nguyen2025coarsehickingbotham2025graphs
• Conjecture 2: nguyen2025coarse
• Theorem 3
• Corollary 1
• Proposition 1: dallard2024treewidth
• Proposition 2: DBLP:conf/iwoca/DallardDHMPT24
• Proposition 3: conforti1997universally
• Proposition 4
• Proposition 5
• Definition 1: georgakopoulos2023graph