New algorithms for weighted k-domination and total k-domination problems in proper interval graphs

Abstract

Given a positive integer k, a k-dominating set in a graph G is a set of vertices such that every vertex not in the set has at least k neighbors in the set. A total k-dominating set is a set of vertices such that every vertex of the graph has at least k neighbors in the set. The problems of finding the minimum size of a k-dominating, respectively total k-dominating set, in a given graph, are referred to as k-domination, respectively total k-domination. These generalizations of the classical domination and total domination problems are known to be NP-hard in the class of chordal graphs, and, more specifically, even in the classes of split graphs (both problems) and undirected path graphs (in the case of total k-domination). On the other hand, it follows from previous works by Bui-Xuan et al. (2013) [8] and by Belmonte and Vatshelle (2013) [3] that these two families of problems are solvable in time O(|V(G)|3k+4) in the class of interval graphs. We develop faster algorithms for k-domination and total k-domination in the class of proper interval graphs, by means of reduction to a single shortest path computation in a derived directed acyclic graph with O(|V(G)|2k) nodes and O(|V(G)|4k) arcs. We show that a suitable implementation, which avoids constructing all arcs of the digraph, leads to a running time of O(|V(G)|3k). The algorithms are also applicable to the weighted case.