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Weak signed Roman k-domination in digraphs

Volkmann Lutz

Opuscula Mathematica

2024

Vol. 44, no. 2

285--296

Let k ≥ 1 be an integer, and let D be a finite and simple digraph with vertex set V (D). A weak signed Roman k-dominating function (WSRkDF) on a digraph D is a function f : V (D) → {−1, 1, 2} satisfying the condition that ∑x∈N−[v] f(x) ≥ k for each v ∈ V (D), where N−[v] consists of v and all vertices of D from which arcs go into v. The weight of a WSRkDF f is w(f) = ∑v∈V (D) f(v). The weak signed Roman k-domination number [formula] is the minimum weight of a WSRkDF on D. In this paper we initiate the study of the weak signed Roman k-domination number of digraphs, and we present different bounds on [formula]. In addition, we determine the weak signed Roman k-domination number of some classes of digraphs. Some of our results are extensions of well-known properties of the weak signed Roman domination number [formula] and the signed Roman k-domination number [formula].