Computational Aspects of Double Dominating Sequences in Graphs

Abstract

In a graph $G=(V,E)$, a vertex $u \in V$ dominates a vertex $v \in V$ if $v \in N_G[u]$. A sequence $S=(v_1,v_2, \ldots , v_k)$ of vertices of G is called a double dominating sequence of G if (i) for each i, the vertex $v_i$ dominates at least one vertex $u \in V$ which is dominated at most once by the previous vertices of S and, (ii) all vertices of G have been dominated at least twice by the vertices of S. Grundy Double Domination problem asks to find a double dominating sequence of maximum length for a given graph G. In this paper, we prove that the decision version of the problem is NP-complete for bipartite and co-bipartite graphs. We look for the complexity status of the problem in the class of chain graphs which is a subclass of bipartite graphs. We use dynamic programming approach to solve this problem in chain graphs and propose an algorithm which outputs a Grundy double dominating sequence of a chain graph G in linear-time.