Recoloring via modular decomposition

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 $\ell \geq χ(G)$+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.

Figures (3)

• Figure 1:
• Figure 2: A graph $G$, its skeleton, and its clique skeleton.
• Figure 3: A graph $G$ (left) and a blowup $G^{'}$ of it (right).

Theorems & Definitions (55)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Lemma 1
• proof
• Theorem 8: Garey1974