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A hierarchy of self-clique graphs. (English) Zbl 1042.05073
Summary: The clique graph \(K(G)\) of \(G\) is the intersection graph of all its (maximal) cliques. A connected graph \(G\) is self-clique whenever \(G \cong K(G)\). Self-clique graphs have been studied in several papers. Here we propose a hierarchy of self-clique graphs: Type 3 \(\subsetneq\) Type 2 \(\subsetneq\) Type 1 \(\subsetneq\) Type 0. We give characterizations for classes of Types 3, 2 and 1 (including Helly self-clique graphs) and several new constructions of families of self-clique graphs. It is shown that all (but one) previously published examples of self-clique graphs are of Type 2. Our methods provide a unified approach and generalizations of those examples. As further applications, we give a characterization of the self-clique graphs such that at most 3 cliques have more than two vertices (they are all of Type 2) and a description of the diamond-free graphs of Type 2.

MSC:
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
Software:
GAP
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References:
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