Let G = (V, E) be a graph of order n. Let B(S) be the set of vertices in V\S that have a neighbor in the vertex set S. The differential of a vertex set S is defined as ∂(S) = |B(S)|−|S| and the maximum value of ∂(S) for any subset S of V is the differential of G. For S ⊆ V(G), the set Np(S) is defined as the perfect neighborhood of S such that all vertices in V(G)\S have exactly one neighbor in S. The perfect differential of S is defined to be ∂p(S) = |Np(S)|−|S| and the perfect differential of a graph is defined as ∂p(G) = {∂p(S)⊆V(G)}. A Roman dominating function of G is a function ƒ:V → {0,1,2} such that every vertex v for which ƒ(v) = 0 has a neighbor u with ƒ(u) = 2. The weight of a Roman dominating function ƒ is w(ƒ) = ∑v∈Vƒ(v). The Roman domination number of a graph G, denoted by γR(G), is the minimum weight of all possible Roman dominating functions. A perfect Roman dominating function is defined as an Roman dominating function ƒ satisfying the condition that every vertex u for which ƒ(u) = 0 is adjacent to exactly one vertex v for which ƒ(v) = 2. The perfect Roman domination number, denoted by γPR(G), is the minimum weight among all perfect Roman dominating functions on G, that is γPR(G) = min{w(ƒ):ƒ is a perfect Roman dominating function on G}. This paper is devoted to the computation of differential, perfect differential and Roman domination, perfect Roman domination of probabilistic neural networks by the use of the proven Gallai-type results γR(G) = n−∂(G), γPR(G) + ∂P(G) = n. Besides, existing Roman and perfect Roman graph classes of probabilistic neural networks are characterized.
2020 Mathematics Subject Classification: 05C69, 05C05, 05C82