Recognizing graphs close to bipartite graphs with an application to colouring reconfiguration
Marthe Bonamy, Konrad K. Dabrowski, Carl Feghali, Matthew Johnson, Daniël Paulusma · Journal of Graph Theory · 2021
Abstract We continue research into a well‐studied family of problems that ask whether the vertices of a given graph can be partitioned into sets and , where is an independent set and induces a graph from some specified graph class . We consider the case where is the class of ‐degenerate graphs. This problem is known to be polynomial‐time solvable if (recognition of bipartite graphs), but ‐complete if (near‐bipartite graphs) even for graphs of maximum degree 4. Yang and Yuan showed that the case is polynomial‐time solvable for graphs of maximum degree 3. This also follows from a result of Catlin and Lai. We study the general case for ‐vertex graphs of maximum degree . We show how to find and in time for , and in time for . Together, these results provide an algorithmic version of a result of Catlin and also provide an algorithmic version of a generalization of Brook's Theorem, proved by Borodin et al. and Matamala. The results also enable us to solve an open problem of Feghali et al. For a given graph and positive integer , the vertex colouring reconfiguration graph of has as its vertex set the set of ‐colourings of and contains an edge between each pair of colourings that differ on exactly on vertex. We complete the complexity classification of the problem of finding a path in the reconfiguration graph between two given ‐colourings of a given graph of maximum degree .