Let DD be a finite simple digraph with vertex set V(D)V(D) and arc set A(D)A(D). A twin signed Roman dominating function (TSRDF) on the digraph DD is a function f:V(D)→{−1,1,2}f:V(D)→{−1,1,2} satisfying the conditions that (i) ∑x∈N−[v]f(x)≥1∑x∈N−[v]f(x)≥1 and ∑x∈N+[v]f(x)≥1∑x∈N+[v]f(x)≥1 for each v∈V(D)v∈V(D), where N−[v]N−[v] (resp. N+[v]N+[v]) consists of vv and all in-neighbors (resp. out-neighbors) of vv, and (ii) every vertex uu for which f(u)=−1f(u)=−1 has an in-neighbor vv and an out-neighbor ww for which f(v)=f(w)=2f(v)=f(w)=2. The weight of an TSRDF ff is ω(f)=∑v∈V(D)f(v)ω(f)=∑v∈V(D)f(v). The twin signed Roman domination number γ∗sR(D)γsR∗(D) of DD is the minimum weight of an TSRDF on DD. In this paper, we initiate the study of twin signed Roman domination in digraphs and we present some sharp bounds on γ∗sR(D)γsR∗(D). In addition, we determine the twin signed Roman domination number of some classes of digraphs.
