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

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Algorithms and Complexity (CIAC 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9079))

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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.

The research in this paper was supported by EPSRC (EP/G043434/1 and EP/K025090/1) and ANR (TODO ANR-09-EMER-010).

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Authors and Affiliations

  1. School of Engineering and Computing Sciences, Durham University Science Laboratories, South Road, Durham,  DH1 3LE, UK

    Konrad K. Dabrowski & Daniël Paulusma

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  1. Konrad K. Dabrowski
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  2. Daniël Paulusma
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Correspondence to Konrad K. Dabrowski .

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  1. LAMSADE, Université Paris-Dauphine, Paris Cedex 16, France

    Vangelis Th. Paschos

  2. Inst. of Theoretical Computer Science, ETH Zürich, Zürich, Switzerland

    Peter Widmayer

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Dabrowski, K.K., Paulusma, D. (2015). Clique-Width of Graph Classes Defined by Two Forbidden Induced Subgraphs. In: Paschos, V., Widmayer, P. (eds) Algorithms and Complexity. CIAC 2015. Lecture Notes in Computer Science(), vol 9079. Springer, Cham. https://doi.org/10.1007/978-3-319-18173-8_12

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  • DOI: https://doi.org/10.1007/978-3-319-18173-8_12

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