On the isolated signed (edge) domination number of graphs

Abstract

An isolated signed dominating function (ISDF) on a graph $G$ is a signed dominating function whose closed neighborhood sum is equal to $1$ for at least one vertex. The minimum weight of such a function is called the isolated signed domination number, denoted by $\gamma_{is}(G)$. This notion was introduced by Dayana and Kumar.}   {In this paper, we continue the study of $\gamma_{is}(G)$ by establishing several new bounds and properties. We also introduce the isolated signed edge domination number $\gamma'_{is}(G)$ through its corresponding edge version of signed domination. Fundamental properties of this parameter are investigated, upper and lower bounds are obtained, exact values are determined for several families of graphs, and the associated decision problem is shown to be NP-complete for bipartite graphs.