R-domination Problems on Homogeneously Orderable Graphs

In this paper, we consider r-dominating cliques in homogeneously orderable graphs (a common generalization of dually chordal and distance-hereditary graphs) and their relation to strict r-packing sets. We prove that a homogeneously orderable graph G possesses an r-dominating clique if and only if for any pair of vertices x , y of G d (x , y ) ° r ( x ) / r ( y ) / 1 holds where r : V r N is a given vertex function. Furthermore, we show that for homogeneously orderable graphs with r-dominating cliques the cardinality of a maximum strict r-packing set equals the cardinality of a minimum r-dominating clique provided the last parameter is not two. Finally, we present two efficient algorithms: The first one decides whether a given homogeneously orderable graph has an r-dominating clique and, if so, computes both a minimum r-dominating clique and a maximum strict r-packing set of the graph. The second one computes a minimum connected r-dominating set in a homogeneously orderable graph. q 1997 John Wiley & Sons, Inc. Networks 30: 121–131, 1997