TWIN SIGNED ROMAN DOMATIC NUMBERS IN DIGRAPHS

SEYED MAHMOUD SHEIKHOLESLAMI ； LUTZ VOLKMANN

《Tamkang Journal of Mathematics》 48卷3期 (2017/09) Pp. 265-272

https://doi.org/10.5556/j.tkjm.48.2017.2306

摘要

Let D be a finite simple digraph with vertex set V(D). A twin signed Roman dominating function on the digraph D is a function f: V(D) → {-1, 1, 2} satisfying the conditions that (i) Σ_(x∈N^-[v]) f(x) ≥ 1 and Σ_(x∈N^+[v]) f(x) ≥ 1 for each v ∈ V(D), where N^-[v] (resp. N^+[v]) consists of v and all in-neighbors (resp. out-neighbors) of v, and (ii) every vertex u for which f(u) = -1 has an in-neighbor v and an out-neighbor w for which f(v) = f(w) = 2. A set {f_1, f_2, ..., f_d } of distinct twin signed Roman dominating functions on D with the property that Σ_(i=1)^d f_i(v) ≤ 1 for each v ∈ V(D), is called a twin signed Roman dominating family (of functions) on D. The maximum number of functions in a twin signed Roman dominating family on D is the twin signed Roman domatic number of D, denoted by d_(sR)^*(D). In this paper, we initiate the study of the twin signed Roman domatic number in digraphs and we present some sharp bounds on d_(sR)^*(D). In addition, we determine the twin signed Roman domatic number of some classes of digraphs.