Clique r-Domination and Clique r-Packing Problems on Dually Chordal Graphs

Let $\cal C$ be a family of cliques of a graph G=(V,E). Suppose that each clique C of $\cal C$ is associated with an integer r(C)$, where $r(C) \ge 0$. A vertex v r-dominates a clique C of G if $d(v,x) \le r(C)$ for all $x \in C$, where d(v,x) is the standard graph distance. A subset $D \subseteq V$ is a clique r-dominating set of G if for every clique $C \in \cal C$ there is a vertex $u \in D$ which r-dominates C. A clique r-packing set is a subset $P \subseteq \cal C$ such that there are no two distinct cliques $C',C''\in P$ $r$-dominated by a common vertex of G. The clique r-domination problem is to find a clique r-dominating set with minimum size and the clique r-packing problem is to find a clique r-packing set with maximum size. The formulated problems include many domination and clique-transversal-related problems as special cases. In this paper an efficient algorithm is proposed for solving these problems on dually chordal graphs which are a natural generalization of strongly chordal graphs. The efficient algorithm is mainly based on the tree structure and special vertex elimination orderings of dually chordal graphs. In some important particular cases where the algorithm works in linear time the obtained results generalize and improve known results on strongly chordal graphs.