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arXiv · 2503.00798

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

Dibyayan Chakraborty
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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)].

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BibTeXRIS

Dibyayan Chakraborty. 2025-03-02. $K_{2,3}$-induced minor-free graphs admit quasi-isometry with additive distortion to graphs of tree-width at most two. https://arxiv.org/abs/2503.00798

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