Inverse isolated signed domination function of digraphs

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)→{−1,+1} if ∑
x∈N+[v]
f (x) ≤0forevery v∈V(D)
and for at least one vertex w ∈V(D), f(N+[w]) =0. The notation γ0
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 inverse isolated signed dominating functions and
finding the precise values of γ0
is(G) for some families of graphs.