Inverse isolated signed domination function of digraphs

Abstract

Let $D = (V, A)$ be a directed graph with $q$ arcs and $p$ vertices. The inverse isolated signed dominating function (IISDF) is a function $f: V(G) \rightarrow \{-1,+1\}$ if $\sum\limits_{x\in N^{+}[v]}f(x) \leq 0$ for every $v \in V(D)$ and for at least one vertex $w \in V(D), f(N^{+}[w]) = 0$. The notation $\gamma^{0}_{is}(D)$, which is the greatest weight of an IISDF of $D$, represents the inverse isolated signed dominance number (IISDN) for a digraph. In this work, we provide foundational results and characterizations for this recently proposed form of signed domination by proving the existence of  and finding the precise values of $\gamma_{is}^{0}(G)$ for some families of graphs.