Clique-width of graph classes defined by two forbidden induced subgraphs

Dabrowski, K. K.; Paulusma, D.

Abstract

If a graph has no induced subgraph isomorphic to any graph in a finite family {H1,…,Hp}, it is said to be (H1,…,Hp)-free. The class of H-free graphs has bounded clique-width if and only if H is an induced subgraph of the 4-vertex path P4. We study the (un)boundedness of the clique-width of graph classes defined by two forbidden induced subgraphs H1 and H2. Prior to our study it was not known whether the number of open cases was finite. We provide a positive answer to this question. To reduce the number of open cases we determine new graph classes of bounded clique-width and new graph classes of unbounded clique-width. For obtaining the latter results we first present a new, generic construction for graph classes of unbounded clique-width. Our results settle the boundedness or unboundedness of the clique-width of the class of (H1,H2)-free graphs for all pairs (H1,H2), both of which are connected, except two non-equivalent cases, and for all pairs (H1,H2), at least one of which is not connected, except 11 non-equivalent cases. We also consider classes characterized by forbidding a finite family of graphs {H1,…,Hp} as subgraphs, minors and topological minors, respectively, and completely determine which of these classes have bounded clique-width. Finally, we show algorithmic consequences of our results for the graph colouring problem restricted to (H1,H2)-free graphs.

Citation

Dabrowski, K. K., & Paulusma, D. (2015, March). Clique-width of graph classes defined by two forbidden induced subgraphs. Presented at 9th International Conference on Algorithms and Complexity (CIAC 2015), Paris, France

• Presentation Conference Type: Conference Paper (published)
• Conference Name: 9th International Conference on Algorithms and Complexity (CIAC 2015)
• Start Date: Mar 20, 2015
• End Date: Mar 22, 2015
• Publication Date: May 16, 2015
• Deposit Date: Dec 22, 2014
• Publicly Available Date: May 16, 2016
• Print ISSN: 0302-9743
• Pages: 167-181
• Series Title: Lecture notes in computer science
• Series Number: 9079
• Series ISSN: 0302-9743
• Book Title: Algorithms and complexity : 9th International Conference, CIAC 2015, Paris, France, May 20-22, 2015 ; proceedings.
• ISBN: 9783319181721
• DOI: https://doi.org/10.1007/978-3-319-18173-8_12
• Keywords: Clique-width, Forbidden induced subgraph, Graph class
• Public URL: https://durham-repository.worktribe.com/output/1153372