$K_{2,3}$-induced minor-free graphs admit quasi-isometry with additive distortion to graphs of tree-width at most two
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2025
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| author | Chakraborty, Dibyayan |
| author_facet | Chakraborty, Dibyayan |
| contents | 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)]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_00798 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $K_{2,3}$-induced minor-free graphs admit quasi-isometry with additive distortion to graphs of tree-width at most two Chakraborty, Dibyayan Combinatorics Discrete Mathematics 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)]. |
| title | $K_{2,3}$-induced minor-free graphs admit quasi-isometry with additive distortion to graphs of tree-width at most two |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2503.00798 |