Weak Signed Roman Domination in Digraphs

LetD be a finite and simple digraph with vertex set V (D). A weak signed Roman dominating function (WSRDF) on a digraphD is a function f : V (D) → {−1, 1, 2} satisfying the condition that ∑ x∈N−[v] f(x) ≥ 1 for each v ∈ V (D), whereN−[v] consists of v and all vertices ofD from which arcs go into v. The weight of a WSRDF f is ∑ v∈V (D) f(v). The weak signed Roman domination number γwsR(D) of D is the minimum weight of a WSRDF on D. In this paper we initiate the study of the weak signed Roman domination number of digraphs, and we present different bounds on γwsR(D). In addition, we determine the weak signed Roman domination number of some classes of digraphs. 1 Terminology and introduction In this paper we continue the study of Roman dominating functions in graphs and digraphs. Let G be a simple graph with vertex set V (G), and letN [v] = NG[v] be the closed neighborhood of the vertex v. A signedRomandominating functionon a graphG is defined in [1] as a function f : V (G) −→ {−1, 1, 2} satisfying the conditions that (i) f(NG[v]) = ∑ x∈NG[v] f(x) ≥ 1 for every v ∈ V (G), and (ii) for every vertex u for which f(u) = −1 is adjacent to a vertex v for which f(v) = 2. The weight of a signed Roman dominating function f on a graphG is ∑ v∈V (G) f(v). The signed Roman domination number γsR(G) ofG is the minimum weight of a signed Roman dominating function onG. A weak signed Roman dominating function on a graph G is defined in [14] as a function f : V (G) −→ {−1, 1, 2} such that f(NG[v]) ≥ 1 for every v ∈ V (G). The weight of a weak signed Roman dominating function f on a graph G is ∑ v∈V (G) f(v). The weak signed Roman domination number γwsR(G) of G is the minimum weight of a weak signed Roman dominating function onG. Clearly, γwsR(G) ≤ γsR(G).