Clique-Width of Graph Classes Defined by Two Forbidden Induced Subgraphs

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9079)

Abstract

If a graph has no induced subgraph isomorphic to any graph in a finite family \(\{H_1,\ldots ,H_p\}\), it is said to be \((H_1,\ldots ,H_p)\)-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 \(P_4\). We study the (un)boundedness of the clique-width of graph classes defined by two forbidden induced subgraphs \(H_1\) and \(H_2\). 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 \((H_1,H_2)\)-free graphs
  1. (i)

    for all pairs \((H_1,H_2)\), both of which are connected, except two non-equivalent cases, and

     
  2. (ii)

    for all pairs \((H_1,H_2)\), 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 \(\{H_1,\ldots ,H_p\}\) 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 \((H_1,H_2)\)-free graphs.

Keywords

Clique-width Forbidden induced subgraph Graph class 

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.School of Engineering and Computing SciencesDurham University Science LaboratoriesDurhamUK

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