Disjoint dominating and 2-dominating sets in graphs

Abstract

A subset D⊆VG is a dominating set of G if every vertex in VG−D has a neighbor in D, while D is 2-dominating if every vertex in VG−D has at least two neighbors in D. A graph G is a DD2-graph if it has a pair (D,D2) of disjoint subsets of vertices such that D is a dominating set, and D2 is a 2-dominating set of G. Studies of first properties of the DD2-graphs were initiated by Henning and Rall (2013). In this paper, we continue their study and complete their structural characterization of all DD2-graphs. Next, we focus on minimal DD2-graphs and provide the relevant characterization of that class of graphs as well. Additionally, we study optimization problems related to DD2-graphs and non-DD2-graphs, respectively. In particular, for a given DD2-graph G, the purpose is to find a minimal spanning DD2-graph of G of minimum or maximum size. We show that both these problems are NP-hard. Finally, if G is a graph which is not a DD2-graph, we consider the question of how many edges must be added to G or subdivided in G to ensure the existence of a DD2-pair in the resulting graph. The latter problem turned out to be polynomially tractable, while the former one is NP-hard.