Twin signed Roman domatic numbers in digraphs

Authors

  • Seyed Mahmoud Sheikholeslami
  • Lutz Volkmann

DOI:

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

Keywords:

twin signed Roman dominating function, twin signed Roman domination number, twin signed Roman domatic number, directed graph

Abstract

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 $f:V(D)\rightarrow\{-1,1,2\}$ 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 $v$ 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 $\{f_1,f_2,\ldots,f_d\}$ of distinct twin signed 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 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.

Author Biographies

Seyed Mahmoud Sheikholeslami

Department ofMathematics, Azarbaijan ShahidMadani University, Tabriz, I.R. Iran.

Lutz Volkmann

Lehrstuhl II fürMathematik, RWTH Aachen University, 52056 Aachen, Germany.

References

H. Abdollahzadeh Ahangar, M. A. Henning, Y. Zhao, C. Lowenstein and V. Samodivkin, Signed Roman domination in graphs, J. Comb. Optim., 27(2014), 241--255.

A. Bodaghli, S. M. Sheikholeslami and L. Volkmann, Twin signed Roman domination numbers in directed graphs, Tamkang J. Math., 47(2016), 357--371.

M. A. Henning and L. Volkmann, Signed Roman $k$-domination in graphs, Graphs Combin., 32(2016), 175--190.

S. M. Sheikholeslami and L. Volkmann, The signed Roman domatic number of a graph, Ann. Math. Inform., 40(2012), 105--112.

S. M. Sheikholeslami and L. Volkmann, Signed Roman domination in digraphs, J. Comb. Optim., 30(2015), 456--467.

S. M. Sheikholeslami and L. Volkmann, The signed Roman

domatic number of a digraph, Electro. J. Graph Theory Appl. 3(2015), 85--93.

D. B. West, Introduction to Graph Theory, Prentice-Hall, Inc, 2000.

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Published

2017-09-30

How to Cite

Sheikholeslami, S. M., & Volkmann, L. (2017). Twin signed Roman domatic numbers in digraphs. Tamkang Journal of Mathematics, 48(3), 265–272. https://doi.org/10.5556/j.tkjm.48.2017.2306

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Papers

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