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Diameters of iterated clique graphs of chordal graphs. (English) Zbl 0726.05059
The clique graph K(G) of a graph G is the intersection graph of the cliques (i.e. the maximal complete subgraphs) of G. Iterated clique graphs \(K^ n(G)\) are defined as usual by \(K(K^{n-1}(G))\) for \(n\geq 2\), where \(K^ 1(G):=K(G)\). B. Hedman [J. Comb. Theory, Ser. B 37, 270-278 (1984; Zbl 0547.05056)] had shown that the difference of the diameters of K(G) and \(K^ n(G)\) equals n if G is a connected unit interval graph of diameter at least n. The main result of the present paper states this being true for connected chordal graphs also. Recall that a graph is chordal if every induced cycle is a triangle. A tool, but interesting for its own sake, is the following result: \(K^ 2(G)\), but not necessarily K(G), is chordal for any chordal graph G. Independently these results were also obtained by H. J. Bandelt and E. Prisner [J. Comb. Theory, Ser. B 51, No.1, 34-45 (1991; see following review)].

MSC:
05C99 Graph theory
05C38 Paths and cycles
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References:
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