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)→{−1,1,2} satisfying the conditions that (i) ∑x∈N−(v)f(x)≥1 and ∑x∈N+(v)f(x)≥1 for each v∈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 {f1,f2,…,fd} of distinct twin signed total Roman dominating functions on D with the property that ∑i=1dfi(v)≤1 for each v∈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 dstR∗(D). In this paper, we initiate the study of the twin signed total Roman domatic number in digraphs and present some sharp bounds on dstR∗(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.
 
Communications in Combinatorics and Optimization
Volume 6, Issue 1
June 2021
Pages 17-26

Files

Share

How to cite

Statistics