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Abstract
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Abstract. Let k ≥ 1 be an integer and G be a simple and finite graph with vertex set
V (G). A signed Roman k-dominating function (SRkDF) on a graph G is a function
f : V (G) → {−1, 1, 2} such that (i) every vertex v with f(v) = −1 is adjacent to at
least one vertex u with f(u) = 2, (ii) P
u∈N[v]
f(u) ≥ k holds for any vertex v. The
weight of a SRkDF f is P
u∈V (G)
f(u), and the minimum weight of a SRkDF is the
signed Roman k-domination number γ
k
sR(G) of G. In this paper, we investigate the
signed Roman k-domination number of graphs, and we establish some bounds on
γ
k
sR(G). In the case that T is a tree, we present lower and upper bounds on γ
k
sR(T)
for k ∈ {3, 4} and classify all extremal trees.
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