Building a larger class of graphs for efficient reconfiguration of vertex colouring

Abstract

A $k$-colouring of a graph $G$ is an assignment of at most $k$ colours to the vertices of $G$ so that adjacent vertices are assigned different colours. The reconfiguration graph of the $k$-colourings, $\mathcal{R}_k(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two colourings are joined by an edge in $\mathcal{R}_k(G)$ if they differ in colour on exactly one vertex. For a $k$-colourable graph $G$, we investigate the connectivity and diameter of $\mathcal{R}_{k+1}(G)$. It is known that not all weakly chordal graphs have the property that $\mathcal{R}_{k+1}(G)$ is connected. On the other hand, $\mathcal{R}_{k+1}(G)$ is connected and of diameter $O(n^2)$ for several subclasses of weakly chordal graphs such as chordal, chordal bipartite, and $P_4$-free graphs.

We introduce a new class of graphs called OAT graphs that extends the latter classes and in fact extends outside the class of weakly chordal graphs. OAT graphs are built from four simple operations, disjoint union, join, and the addition of a clique or comparable vertex. We prove that if $G$ is a $k$-colourable OAT graph, then $\mathcal{R}_{k+1}(G)$ is connected with diameter $O(n^2)$. Furthermore, we give polynomial time algorithms to recognize OAT graphs and to find a path between any two colourings in $\mathcal{R}_{k+1}(G)$.

Feghali and Fiala defined a subclass of weakly chordal graphs, called compact graphs, and proved that for every $k$-colourable compact graph $G$, $\mathcal{R}_{k+1}(G)$ is connected with diameter $O(n^2)$. We prove that the class of OAT graphs properly contains the class of compact graphs. Feghali and Fiala also asked if for a $k$-colourable ($P_5$, co-$P_5$, $C_5$)-free graph $G$, $\mathcal{R}_{k+1}(G)$ is connected with diameter $O(n^2)$. We answer this question in the positive for the subclass of $P_4$-sparse graphs, which are the ($P_5$, co-$P_5$, $C_5$, $P$, co-$P$, fork, co-fork)-free graphs.