On clique convergent graphs
The clique graph \(K(G)\) of a graph \(G\) is defined as an intersection graph of the maximal cliques of \(G\). The authors investigate graphs for which an integer \(n\) exists such that \(K^n(G)\) is a trivial graph, where \(K^n(G)\) is the \(n\)th clique iterated graph of \(G\). The smallest integer \(n\) with the above property is called the index of \(G\). The cliques of \(G\) fulfill the Helly property if every family of pairwise intersecting cliques has nonempty intersection. The Helley defect is defined as the minimal integer \(n\) such that \(K^n(G)\) has the Helly property. The main result of the paper is that for arbitrary integers \(n\) there exists a graph \(G\) such that the Helly defect of \(G\) may exceed by \(n\) the difference of its index and diameter.
scientific article; zbMATH DE number 811341