Building a larger class of graphs for efficient reconfiguration of vertex colouring
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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, R_k(G), is the graph whose vertices are the k-colourings of G and two colourings are joined by an edge in 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 R_k+1(G). It is known that not all weakly chordal graphs have the property that R_k+1(G) is connected. On the other hand, 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 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 R_k+1(G).
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