On the Mixed Minus Domination in Graphs

Abstract

Let G=(V,E) be a graph, for an element x∈V∪E, the open total neighborhood of x is denoted by Nt(x)={y|y is adjacent to x or y is incident with x,y∈V∪E}, and Nt[x]=Nt(x)∪{x} is the closed one. A function f:V(G)∪E(G)→{−1,0,1} is said to be a mixed minus domination function (TMDF) of G if \(\sum_{y \in N_{t}[x]} f(y) \geqslant 1\) holds for all x∈V(G)∪E(G). The mixed minus domination number \(\gamma '_{tm}(G)\) of G is defined as $$\gamma '_{tm}(G) = \min \biggl\{ \sum _{x \in V \cup E} f(x)| f \mbox{is a TMDF of} G \biggr\}. $$