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Durham University

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Publication details

Bonamy, M., Dabrowski, K.K., Feghali, C., Johnson, M. & Paulusma, D. (2017), Recognizing Graphs Close to Bipartite Graphs, in Larsen, Kim G., Bodlaender, Hans L. & Raskin, Jean-Francois eds, LIPIcs–Leibniz International Proceedings in Informatics 83: 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Aalborg, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 70.

Author(s) from Durham


We continue research into a well-studied family of problems that ask if the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a graph from some specified graph class G. We let G be the class of k-degenerate graphs. The problem is known to be polynomial-time solvable if k=0 (bipartite graphs) and NP-complete if k=1 (near-bipartite graphs) even for graphs of diameter 4, as shown by Yang and Yuan, who also proved polynomial-time solvability for graphs of diameter 2. We show that recognizing near-bipartite graphs of diameter 3 is NP-complete resolving their open problem. To answer another open problem, we consider graphs of maximum degree D on n vertices. We show how to find A and B in O(n) time for k=1 and D=3, and in O(n^2) time for k >= 2 and D >= 4. These results also provide an algorithmic version of a result of Catlin [JCTB, 1979] and enable us to complete the complexity classification of another problem: finding a path in the vertex colouring reconfiguration graph between two given k-colourings of a graph of bounded maximum degree.