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 f├╝rMathematik, 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.

Downloads

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

Issue

Section

Papers

Most read articles by the same author(s)