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, also known as a k-tuple total 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 recent work of Kang et al. (2017) that these two families of problems are solvable in time O(|V (G)| 6k+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.

Total Roman domination for proper interval graphs

“…Liu and Chang [11] showed that the decision problem related to total Roman domination number is NP-hard even when restricted to bipartite graphs and chordal graphs. Many authors proposed algorithms to compute some variants of domination on proper interval graphs, a well known subclass of chordal graphs, for example, [6,7,8,13]. In this paper we propose a linear algorithm to compute the total Roman domination number of proper interval graphs.…”

k-tuple and k-tuple total dominations on web graphs

“…For proper interval graphs, an efficient algorithm for the 1-tuple domination problem is de-veloped in [5], but no valid algorithm seems to be currently available for the remaining values of k beyond the one given in [12] for strongly chordal graphs (which is a superclass of proper interval graphs). With a different approach, a polynomial time algorithm was presented recently in [6] for the k-tuple total domination problem in proper interval graphs, for each fixed value of k.…”