r-Dominating cliques in graphs with hypertree structure

Let G = (V, E) be an undirected graph and r be a vertex weight function with positive integer values. A subset (clique) D ~_ V is an r-dominat ing set (clique) in G ifffor every vertex v e V there is a vertex u e D with dist(u, v) <~ r(v). This paper contains the following results: (i) We give a simple necessary and sufficient condition for the existence of r-dominating cliques in the case of Helly graphs and of chordal graphs. (ii) For Helly graphs with an r-dominating clique the minimum size of an r-dominating clique coincides with the minimum size of any r-dominating set. (iii) We give a linear-time algorithm for finding a minimum r-dominating clique in dually chordal graphs (a generalization of strongly chordal graphs). These results improve and extend earlier results on r-dominating cliques in chordal and strongly chordal graphs.