Signed Edge \(k\)-Subdomination Numbers in Graphs

Abstract

The closed neighborhood \(N_G[e]\) of an edge \(e\) in a graph \(G\) is the set consisting of \(ev\) and of all edges having a common end-vertex with \(e\). Let \(f\) be a function on \(E(G)\), the edge set of \(G\), into the set \(\{-1, 1\}\). If \(\sum_{x\in E(G)}f(x \geq 1\) for at least \(k\) edges \(e\) of \(G\), then \(f\) is called a signed edge \(k\)-subdominating function of \(G\). The minimum of the values \(\sum_{e \in E(G)} f(e)\), taken over all signed edge \(k\)-subdominating functions \(f\) of \(G\), is called the signed edge \(k\)-subdomination number of \(G\) and is denoted by \(\gamma_{s,k}(G)\). In this note, we initiate the study of the signed edge \(k\)-subdomination in graphs and present some (sharp) bounds for this parameter.