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Abstract
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This paper is devoted to the study of the quadruple Roman domination in trees, and it
is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of
Symmetry. For any positive integer k, a [k]-Roman dominating function ([k]-RDF) of a simple graph
G is a function from the vertex set V of G to the set f0, 1, 2, . . . , k + 1g if for any vertex u 2 V with
f (u) < k, åx2N(u)[fug f (x) jfx 2 N(u) : f (x) 1gj + k, where N(u) is the open neighborhood
of u. The weight of a [k]-RDF is the value Sv2V f (v). The minimum weight of a [k]-RDF is called
the [k]-Roman domination number g[kR](G) of G. In this paper, we establish sharp upper and lower
bounds on g[4R](T) for nontrivial trees T and characterize extremal trees.
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