Bounding Clique-Width via Perfect Graphs
Given two graphs \(H_1\) and \(H_2\), a graph \(G\) is \((H_1,H_2)\)-free if it contains no subgraph isomorphic to \(H_1\) or \(H_2\). We continue a recent study into the clique-width of \((H_1,H_2)\)-free graphs and present three new classes of \((H_1,H_2)\)-free graphs that have bounded clique-width. We also show the implications of our results for the computational complexity of the Colouring problem restricted to \((H_1,H_2)\)-free graphs. The three new graph classes have in common that one of their two forbidden induced subgraphs is the diamond (the graph obtained from a clique on four vertices by deleting one edge). To prove boundedness of their clique-width we develop a technique based on bounding clique covering number in combination with reduction to subclasses of perfect graphs.
KeywordsClique-width Forbidden induced subgraphs Graph class
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