On the signed Roman edge k-domination in graphs

Document Type : Original paper


Department of Mathematics Payame Noor University I.R. Iran


Let $kgeq 1$ be an integer, and $G=(V,E)$ be a finite and simple
graph. The closed neighborhood $N_G[e]$ of an edge $e$ in a graph
$G$ is the set consisting of $e$ and all edges having a common
end-vertex with $e$. A signed Roman edge $k$-dominating function
(SREkDF) on a graph $G$ is a function $f:E rightarrow
{-1,1,2}$ satisfying the conditions that (i) for every edge $e$
of $G$, $sum _{xin N[e]} f(x)geq k$ and (ii) every edge $e$
for which $f(e)=-1$ is adjacent to at least one edge $e'$ for
which $f(e')=2$. The minimum of the values $sum_{ein E}f(e)$,
taken over all signed Roman edge $k$-dominating functions $f$ of
$G$, is called the signed Roman edge $k$-domination number of $G$
and is denoted by $gamma'_{sRk}(G)$. In this paper we establish some new bounds on the signed Roman edge $k$-domination number.


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