Inverse isolated signed domination function of graphs

Abstract

Let $G$ be a graph of order $p$ and size $q$. The function $f: V(G) \rightarrow \{-1,+1\}$ is the inverse isolated signed dominating function (IISDF) if $\sum\limits_{x\in N[v]}f(x) \leq 0$ for every $v \in V(G)$ and for at least one vertex $w \in V(G), f(N[w]) = 0$.  The inverse isolated signed dominating number (IISDN) for a graph is represented by the notation $\gamma^{0}_{is}(G)$, which is the greatest weight of an IISDF of $G$. In this work, we study IISDF for some graph families.