INVERSE MINUS DOMINATION IN GRAPHS

Abstract

A minus dominating function on a graph $G=(V, E)$ is a labelling $f: V \rightarrow\{-1,0,1\}$ such that $f(N[v]) \geq 1$ for all $v \in V$. The sum of all labellings of $f$ is the weight of $f$ denoted by $w(f)$. If $f$ is a minus dominating function on $G$ with minimum weight, then its inverse minus dominating function $f^{\prime}$ is a minus dominating function on $G$ such that $f^{\prime}(v) \neq 1, \quad \forall v \in S$, where $S=\{v \in V / f(v)=1\}$. The minimum weight of such a function is the inverse minus domination number of $G$ denoted by $\gamma_{i m}(G)$. In this paper, we initiate this new type of domination, obtain some properties and characterize the graphs attaining bounds.

Received: May 17, 2021
Revised: June 3, 2021
Accepted: November 14, 2021