Faster Algorithms for the Paired Domination Problem on Interval and Cir ular-Ar Graphs

Abstra t A vertex subset D of a graph G is a dominating set if every vertex of G is either in D or is adja ent to a vertex in D. The paired domination problem on G asks for a minimumardinality dominating set S of G su h that the subgraph indu ed by S ontains a perfe t mat hing; motivation for this problem omes from the interest in nding a small number of lo ations to pla e pairs of mutually visible guards so that the entire set of guards monitors a given area. The paired domination problem on general graphs is known to be NPomplete. In this paper, we onsider the paired domination problem on interval and ir ular-ar graphs. We use properties of the models of interval and ir ular-ar graphs in order to des ribe simple and eÆ ient algorithms for the problem: given an interval (ar , resp.) model of an interval ( ir ular-ar , resp.) graph on n verti es and m edges with endpoints sorted, our algorithms dete t whether there exist isolated verti es, returning one if one exists, otherwise returning a minimum paired-dominating set of the input graph; our algorithm for interval graphs runs in O(n) time and spa e whereas the one for ir ular ar graphs runs in O(n + m) time using O(n) spa e. Both algorithms a hieve better time omplexities over the orresponding known algorithms.