On the Signed Edge Domination Numbers of \(K_{m,n}\)

Abstract

Let \(G = (V, E)\) be a simple undirected graph. For an edge \(e\) of \(G\), the \({closed\; edge-neighborhood}\) of \(e\) is the set \(N[e] = \{e’ \in E \mid e’ \text{ is adjacent to } e\} \cup \{e\}\). A function \(f: E \to \{1, -1\}\) is called a signed edge domination function (SEDF) of \(G\) if \(\sum_{e’ \in N[e]} f(e’) > 1\) for every edge \(e\) of \(G\). The signed edge domination number of \(G\) is defined as \(\gamma’_s(G) = \min \left\{ \sum_{e \in E} |f(e)| \mid f \text{ is an SEDF of } G \right\}\). In this paper, we determine the signed edge domination numbers of all complete bipartite graphs \(K_{m,n}\), and therefore determine the signed domination numbers of \(K_m \times K_n\).