Algorithmic aspects of the $k$-domination problem in graphs

Abstract

For a positive integer $k$, a $k$-dominating set of a graph $G$ is a subset $D\subseteq V\left(G\right)$ such that every vertex not in $D$ is adjacent to at least $k$ vertices in $D$. The $k$-domination problem is to determine a minimum $k$-dominating set of $G$. This paper studies the $k$-domination problem in graphs from an algorithmic point of view. In particular, we present a linear-time algorithm for the $k$-domination problem for graphs in which each block is a clique, a cycle or a complete bipartite graph. This class of graphs includes trees, block graphs, cacti and block-cactus graphs. We also establish NP-completeness of the $k$-domination problem in split graphs.