Unique response strong Roman dominating functions of graphs - Doost Ali Mojdeh; Guoliang Hao; Iman Masoumi; Ali Parsian (Electronic Journal of Graph Theory and Applications (EJGTA), 2021)
Given a simple graph G=(V,E) with maximum degree Δ. Let (V0, V1, V2) be an ordered partition of V, where Vi = {v ∈ V : f(v)=i} for i = 0, 1 and V2 = {v ∈ V : f(v)≥2}. A function f : V → {0, 1, …, ⌈Δ/2⌉+1} is a strong Roman dominating function (StRDF) on G, if every v ∈ V0 has a neighbor w ∈ V2 and f(w)≥1 + ⌈1/2|N(w)∩V0|⌉. A function f : V → {0, 1, …, ⌈Δ/2⌉+1} is a unique response strong Roman function (URStRF), if w ∈ V0, then |N(w)∩V2|≤1 and w ∈ V1 ∪ V2 implies that |N(w)∩V2|=0. A function f : V → {0, 1, …, ⌈Δ/2⌉+1} is a unique response strong Roman dominating function (URStRDF) if it is both URStRF and StRDF. The unique response strong Roman domination number of G, denoted by uStR(G), is the minimum weight of a unique response strong Roman dominating function. In this paper we approach the problem of a Roman domination-type defensive strategy under multiple simultaneous attacks and begin with the study of several mathematical properties of this invariant. We obtain several bounds on such a parameter and give some realizability results for it. Moreover, for any tree T of order n ≥ 3 we prove the sharp bound uStR(T)≤8n/9.