The sub-$k$-domination number of a graph with applications to $k$-domination

Abstract

In this paper we introduce and study a new graph invariant derived from the degree sequence of a graph $G$, called the sub-$k$-domination number and denoted $sub_k(G)$. We show that $sub_k(G)$ is a computationally efficient sharp lower bound on the $k$-domination number of $G$, and improves on several known lower bounds. We also characterize the sub-$k$-domination numbers of several families of graphs, provide structural results on sub-$k$-domination, and explore properties of graphs which are $sub_k(G)$-critical with respect to addition and deletion of vertices and edges.