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

Bonamy, Marthe and Dabrowski, Konrad K. and Feghali, Carl and Johnson, Matthew and Paulusma, DaniΓ«l (2021) 'Recognizing graphs close to bipartite graphs with an application to colouring reconfiguration.', Journal of Graph Theory, 98 (1). pp. 81-109.


We continue research into a well-studied family of problems that ask whether the vertices of a given graph can be partitioned into sets 𝐴 and 𝐡 , where 𝐴 is an independent set and 𝐡 induces a graph from some specified graph class  . We consider the case where  is the class of π‘˜ -degenerate graphs. This problem is known to be polynomial-time solvable if π‘˜=0 (recognition of bipartite graphs), but 𝖭𝖯 -complete if π‘˜=1 (near-bipartite graphs) even for graphs of maximum degree 4. Yang and Yuan showed that the π‘˜=1 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 π‘˜β‰₯1 case for 𝑛 -vertex graphs of maximum degree π‘˜+2 . We show how to find 𝐴 and 𝐡 in 𝑂(𝑛) time for π‘˜=1 , and in 𝑂(𝑛2) time for π‘˜β‰₯2 . 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 𝐺 and positive integer β„“ , the vertex colouring reconfiguration graph of 𝐺 has as its vertex set the set of β„“ -colourings of 𝐺 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 β„“ -colourings of a given graph of maximum degree π‘˜ .

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Publisher statement:This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
Date accepted:03 April 2021
Date deposited:20 August 2021
Date of first online publication:24 May 2021
Date first made open access:20 August 2021

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