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Abstract
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Let D be a finite simple digraph with vertex set V (D). A signed total Roman dominating
function (STRDF) on a digraph D is a function f : V (D) −→ {−1, 1, 2} such that (i)
Pu∈N−(v) f(u) ≥ 1 for every v ∈ V (D), where N−(v) consists of all inner neighbors
of v, and (ii) every vertex u ∈ V (D) for which f(u) = −1 has an inner neighbor v for
which f(v) = 2. The weight of an STRDF f is ω(f) = Σv∈V (D)f(v). The signed total
Roman domination number γstR(D) of D is the minimum weight of an STRDF on D.
A set {f1, f2, . . . , fd} of distinct STRDFs on D with the property that Pd
i=1 fi(v) ≤ 1
for each v ∈ V (D) is called a signed total Roman dominating family (STRD family) (of
functions) on D. The maximum number of functions in an STRD family on D is the
signed total Roman domatic number of D, denoted by dstR(D). In this paper, we initiate
the study of signed total Roman domatic number in digraphs and we present some sharp
bounds for dstR(D). In addition, we determine the signed total Roman domatic number
of some classes of digraphs.
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