Recognizing graphs close to bipartite graphs with an application to colouring reconfiguration

Konrad K Dabrowski,Daniël Paulusma,Matthew Johnson,Marthe Bonamy,Carl Feghali

doi:10.1002/jgt.22683

Abstract

AbstractWe 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 .

Highlights

The COLOURING problem asks if a given graph is k‐colourable for some given integer k; that is, if the vertices of the graph can be coloured with at most k colours, such that no two adjacent vertices are coloured alike
For the sake of example: every planar graph is 5‐degenerate, which implies that it is 6‐colourable, but we can improve this to 5‐colourable by applying the result of Thomassen [46], which states that every planar graph can be decomposed into a 0‐degenerate graph and a 3‐degenerate graph, or by applying another result of Thomassen [45], which states that every planar graph can be decomposed into a 1‐degenerate graph and a 2‐degenerate graph
We proved that recognizing near‐bipartite graphs is NP‐complete for line graphs of maximum degree 4 [5]

Summary

Introduction

The COLOURING problem asks if a given graph is k‐colourable for some given integer k; that is, if the vertices of the graph can be coloured with at most k colours, such that no two adjacent vertices are coloured alike. To prove the latter result, they showed that every connected graph of maximum degree at most 3 is near‐bipartite except K4 (we let Kk denote the complete graph on k vertices).

Results

Conclusion