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arXiv · 2405.06446

Recoloring via modular decomposition

Manoj Belavadi·Kathie Cameron·Ni Luh Dewi Sintiari
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Abstract

The reconfiguration graph of the $k$-colorings of a graph $G$, denoted $R_{k}(G)$, is the graph whose vertices are the $k$-colorings of $G$ and two colorings are adjacent in $R_{k}(G)$ if they differ in color on exactly one vertex. A graph $G$ is said to be recolorable if $R_{\ell}(G)$ is connected for all $\ell \geq \chi(G)$+1. We demonstrate how to use the modular decomposition of a graph class to prove that the graphs in the class are recolorable. In particular, we prove that every ($P_5$, diamond)-free graph, every ($P_5$, house, bull)-free graph, and every ($P_5$, $C_5$, co-fork)-free graph is recolorable. A graph is prime if it cannot be decomposed by modular decomposition except into single vertices. For a prime graph $H$, we study the complexity of deciding if $H$ is $k$-colorable and the complexity of deciding if there exists a path between two given $k$-colorings in $R_{k}(H)$. Suppose $\mathcal{G}$ is a hereditary class of graphs. We prove that if every blowup of every prime graph in $\mathcal{G}$ is recolorable, then every graph in $\mathcal{G}$ is recolorable.

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Manoj Belavadi, Kathie Cameron, Ni Luh Dewi Sintiari. 2024-05-10. Recoloring via modular decomposition. https://doi.org/10.1002/jgt.70005

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