Twin signed total Roman domatic numbers in digraphs

Abstract

Let $D$ be a finite simple digraph with vertex set $V(D)$ and arcset $A(D)$. A twin signed total Roman dominating function (TSTRDF) on thedigraph $D$ is a function $f:V(D)rightarrow{-1,1,2}$ satisfyingthe conditions that (i) $sum_{xin N^-(v)}f(x)ge 1$ and$sum_{xin N^+(v)}f(x)ge 1$ for each $vin V(D)$, where $N^-(v)$(resp. $N^+(v)$) consists of 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 $w$ with$f(v)=f(w)=2$. A set ${f_1,f_2,ldots,f_d}$ of distinct twin signed total Romandominating functions on $D$ with the property that $sum_{i=1}^df_i(v)le 1$for each $vin V(D)$, is called a twin signed total Roman dominating family (offunctions) on $D$. The maximum number of functions in a twin signed total Romandominating family on $D$ is the twin signed total Roman domatic number of $D$,denoted by $d_{stR}^*(D)$. In this paper, we initiate the study of the twinsigned total Roman domatic number in digraphs and present some sharp bounds on$d_{stR}^*(D)$. In addition, we determine the twin signed total Roman domatic numberof some classes of digraphs.