# 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) $\sum_{x\in N^-(v)}f(x)\ge 1$ and $\sum_{x\in N^+(v)}f(x)\ge 1$ for each $v\in V(D)$, 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 $v\in V(D)$, 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 present some sharp bounds on $d_{stR}^*(D)$. In addition, we determine the twin signed total Roman domatic number of some classes of digraphs.

Keywords

Main Subjects

#### References

[1] J Amjadi, The signed total Roman domatic number of a digraph, Discrete Math. Algorithms Appl. 10 (2018), no. 2, ID: 1850020.
[2] J. Amjadi and M. Soroudi, Twin signed total Roman domination numbers in digraphs, Asian-European J. Math. 11 (2018), no. 3, ID: 1850034.
[3] S. Arumugam, K. Ebadi, and L. Sathikala, Twin domination and twin irredundance in digraphs, Appl. Anal. Discrete Math. 7 (2013), no. 2, 275–284.
[4] M. Atapour, A. Bodaghli, and S.M. Sheikholeslami, Twin signed total domination numbers in directed graphs, Ars Combin. 138 (2018), 119–131.
[5] M. Atapour and A. Khodkar, Twin minus domination numbers in directed graphs, Commun. Comb. Optim. 1 (2016), no. 2, 149–164.
[6] M. Atapour, S. Norouzian, S.M. Sheikholeslami, and L. Volkmann, Twin signed domination numbers in directed graphs, Algebra, Discrete Math. 24 (2017), no. 1, 71–89.
[7] J. Bang-Jensen and G.Z. Gutin, Digraphs: Theory, algorithms and applications, Springer Science & Business Media, 2008.
[8] A. Bodaghli, S.M. Sheikholeslami, and L. Volkmann, Twin signed Roman domination numbers in directed graphs, Tamkang J. Math. 47 (2016), no. 3, 357–371.
[9] G. Chartrand, P. Dankelmann, M. Schultz, and H.C. Swart, Twin domination in digraphs, Ars Combin. 67 (2003), 105–114.
[10] N. Dehgardi and M. Atapour, Twin minus total domination numbers in directed graphs, Discuss. Math. Graph Theory 37 (2017), no. 4, 989–1004.
[11] L. Volkmann, On the signed total Roman domination and domatic numbers of graphs, Discrete Appl. Math. 214 (2016), 179–186.
[12] L. Volkmann, Signed total Roman domination in graphs, J. Comb. Optim. 32 (2016), no. 3, 855–871.
[13] L. Volkmann, Signed total Roman domination in digraphs, Discuss. Math. Graph Theory 37 (2017), no. 1, 261–272.