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.

翻译：图$G$的$k$着色重构图，记为$R_{k}(G)$，是以$G$的所有$k$着色为顶点构成的图，其中两个着色在$R_{k}(G)$中相邻当且仅当它们恰在一个顶点上的颜色不同。若对于所有$\ell \geq \chi(G)$+1，$R_{\ell}(G)$都是连通的，则称图$G$是可重新着色的。本文阐述了如何利用图类的模分解来证明该类中的图都是可重新着色的。具体而言，我们证明了每个($P_5$, 菱形)-自由图、每个($P_5$, 屋形图, 牛形图)-自由图以及每个($P_5$, $C_5$, 共叉)-自由图都是可重新着色的。如果一个图无法通过模分解分解为除单顶点以外的结构，则称其为素图。对于素图$H$，我们研究了判定$H$是否$k$可着色以及判定在$R_{k}(H)$中两个给定$k$着色之间是否存在路径的计算复杂度。假设$\mathcal{G}$是一个遗传图类。我们证明，如果$\mathcal{G}$中每个素图的每个膨胀都是可重新着色的，那么$\mathcal{G}$中的每个图都是可重新着色的。