Characterization of classical graph classes by weighted clique graphs.

Given integers m1,...,mℓ, the weighted clique graph of G is the clique graph K(G), in which there is a weight assigned to each complete set S of size mi of K(G), for each i = 1,...,ℓ. This weight equals the cardinality of the intersection of the cliques of G corresponding to S. We characterize weighted clique graphs in similar terms as Roberts and Spencer’s characterization of clique graphs. Further we characterize several classical graph classes in terms of their weighted clique graphs, providing a common framework for describing some different well-known classes of graphs, as hereditary clique-Helly graphs, split graphs, chordal graphs, interval graphs, proper interval graphs, line graphs, among others.