Twin signed total Roman domatic numbers in digraphs

Document Type: Original paper

Author

Azarbaijan

Abstract

Let $D$ be a finite simple digraph with vertex set $V(D)$ and arc
set $A(D)$. A twin signed total Roman dominating function (TSTRDF) on the
digraph $D$ is a function $f:V(D)rightarrow{-1,1,2}$ satisfying
the 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 Roman
dominating 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 (of
functions) on $D$. The maximum number of functions in a twin signed total Roman
dominating 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 twin
signed total Roman domatic number in digraphs and we present some sharp bounds on
$d_{stR}^*(D)$. In addition, we determine the twin signed total Roman domatic number
of some classes of digraphs.

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