Abstract
AbstractA 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 co‐gem are the only two 1‐vertex P4‐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 find four new classes of H‐free chordal graphs of bounded clique‐width. 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 ‐free graphs has bounded clique‐width via a reduction to K4‐free chordal graphs. Finally, we give a complete classification of the (un)boundedness of clique‐width of H‐free weakly chordal graphs.
Highlights
Clique-width is a well-studied graph parameter; see, for example, the surveys of Gurski [40] and Kaminski et al [44]
There are numerous graph classes, such as those that can be characterized by one or more forbidden induced subgraphs,1 for which it has been determined whether or not the class is of bounded clique-width
We study the clique-width of subclasses of chordal graphs, but before going into more detail, we first give some necessary terminology and notation
Summary
Clique-width is a well-studied graph parameter; see, for example, the surveys of Gurski [40] and Kaminski et al [44]. Identifying more graph classes of bounded clique-width and determining what kinds of structural properties ensure that a graph class has bounded clique-width increases our understanding of this parameter. Another important reason for studying these types of questions is that certain classes of NP-complete problems become polynomial-time solvable on any graph class G of bounded clique-width.. Another important reason for studying these types of questions is that certain classes of NP-complete problems become polynomial-time solvable on any graph class G of bounded clique-width.2 Examples of such problems are those definable in Monadic second-order logic using quantifiers on vertices and vertex subsets, but not on edges or edge subsets. We study the clique-width of subclasses of chordal graphs, but before going into more detail, we first give some necessary terminology and notation
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