A Note on the Minus Edge Domination Number in Graphs

Abstract

The closed neighborhood \(N_G[e]\) of an edge \(e\) in a graph \(G\)
is the set consisting of \(e\) 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 \(\{-1, 0, 1\}\). If \(\sum_{x \in N_G[e]} f(x) \geq 1\)
for each \(e \in E(G)\), then \(f\) is called a minus edge
dominating function of \(G\).

The minimum of the values \(\sum_{e \in E(G)} f(e)\), taken over
all minus edge dominating functions \(f\) of \(G\), is called the
\emph{minus edge domination number} of \(G\) and is denoted by
\(\gamma’_m(G)\).

It has been conjectured that \(\gamma’_m(G) \geq n – m\) for every
graph \(G\) of order \(n\) and size \(m\). In this paper, we prove
that this conjecture is true and then classify all graphs \(G\)
with \(\gamma’_m(G) = n – m\).