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 $\sum _{x\in E(G)}f(x\ge 1$ 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.