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arXiv · 2210.14629

Highly unbreakable graph with a fixed excluded minor are almost rigid

Daniel Lokshtanov·Marcin Pilipczuk·Michał Pilipczuk·Saket Saurabh
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Abstract

A set $X \subseteq V(G)$ in a graph $G$ is $(q,k)$-unbreakable if every separation $(A,B)$ of order at most $k$ in $G$ satisfies $|A \cap X| \leq q$ or $|B \cap X| \leq q$. In this paper, we prove the following result: If a graph $G$ excludes a fixed complete graph $K_h$ as a minor and satisfies certain unbreakability guarantees, then $G$ is almost rigid in the following sense: the vertices of $G$ can be partitioned in an isomorphism-invariant way into a part inducing a graph of bounded treewidth and a part that admits a small isomorphism-invariant family of labelings. This result is the key ingredient in the fixed-parameter algorithm for Graph Isomorphism parameterized by the Hadwiger number of the graph, which is presented in a companion paper.

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Daniel Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, Saket Saurabh. 2022-10-26. Highly unbreakable graph with a fixed excluded minor are almost rigid. https://arxiv.org/abs/2210.14629

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