Abstract
A graph is H-free if it has no induced subgraph isomorphic to H. Brandstädt, Engelfriet, Le and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique-width. Brandstädt, Le and Mosca erroneously claimed that the gem and the co-gem are the only two 1-vertex \(P_4\)-extensions H for which the class of H-free chordal graphs has bounded clique-width. In fact we prove that bull-free chordal and co-chair-free chordal graphs have clique-width at most 3 and 4, respectively. In particular, we prove that the clique-width is:
-
(i)
bounded for four classes of H-free chordal graphs;
-
(ii)
unbounded for three subclasses of split graphs.
Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions we determine all graphs H for which the class of H-free chordal graphs has bounded clique-width. We illustrate the usefulness of this classification for classifying other types of graph classes by proving that the class of \((2P_1+ P_3, K_4)\)-free graphs has bounded clique-width via a reduction to \(K_4\)-free chordal graphs. Finally, we give a complete classification of the (un)boundedness of clique-width of H-free weakly chordal graphs.
The research in this paper was supported by EPSRC (EP/K025090/1). The third author is grateful for the generous support of the Graduate (International) Research Travel Award from Simon Fraser University and Dr. Pavol Hell’s NSERC Discovery Grant.
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Notes
- 1.
For a record see also the Information System on Graph Classes and their Inclusions [22].
- 2.
This follows from results [15, 24, 31, 38] that assume the existence of a so-called c-expression of the input graph \(G\in \mathcal{G}\) combined with a result [37] that such a c-expression can be obtained in cubic time for some \(c\le 8^{\hbox {cw}(G)}-1\), where \(\hbox {cw}(G)\) is the clique-width of the graph G.
- 3.
In Theorems 8, 9 and 10, we do not specify our upper bounds as this would complicate our proofs for negligible gain. In our proofs we repeatedly apply graph operations that exponentially increase the upper bound on the clique-width, which means that the bounds that could be obtained from our proofs would be very large and far from being tight. We use different techniques to prove Lemma 5 and Theorem 11, and these allow us to give good bounds for these cases.
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Brandstädt, A., Dabrowski, K.K., Huang, S., Paulusma, D. (2015). Bounding the Clique-Width of H-free Chordal Graphs. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48054-0_12
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