ALGORITHMIC ASPECTS OF MAJORITY DOMINATION

Abstract

This paper studies algorithmic aspects of majority domination, which is a variation of domination in graph theory. A majority dominating function of a graph $G=(V,E)$ is a function $g$ from $V$ to $\{-1,1\}$ such that $\sum_{u\in N[v]}\,g(u)\geq 1$ for at least half of the vertices $v\in V$. The majority domination problem is to find a majority dominating function $g$ of a given graph $G=(V,E)$ such that $\sum_{v\in V}\,g(v)$ is minimized. The concept of majority domination was introduced by Hedetniemi and studied by Broere {\em et al}., who gave exact values for the majority domination numbers of complete graphs, complete bipartite graphs, paths, and unions of two complete graphs. They also proved that the majority domination problem is NP-complete for general graphs; and asked if the problem NP-complete for trees. The main result of this paper is to give polynomial-time algorithms for the majority domination problem in trees, cographs, and $k$-trees with fixed $k$.