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On signed arc total domination in digraphs. (English) Zbl 1403.05109

Summary: Let \(D=(V,A)\) be a finite simple digraph and \(N(uv)=\{u^{\prime}v^{\prime}\neq uv \mid u=u^{\prime}\text{ or }v=v^{\prime}\}\) be the open neighbourhood of \(uv\) in \(D\). A function \(f: A\rightarrow \{-1, +1\}\) is said to be a signed arc total dominating function (SATDF) of \(D\) if \(\sum _{e^{\prime}\in N(uv)}f(e^{\prime})\geq 1\) holds for every arc \(uv\in A\). The signed arc total domination number \(\gamma^{\prime}_{st}(D)\) is defined as \(\gamma^{\prime}_{st}(D)= \min\{\sum_{e\in A}f(e)\mid f \text{ is an SATDF of }D\}\). In this paper we initiate the study of the signed arc total domination in digraphs and present some lower bounds for this parameter.

MSC:

05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
05C20 Directed graphs (digraphs), tournaments
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References:

[1] H. Karami, S.M. Sheikholeslami, A. Khodkar, Lower bounds on signed edge total domina- tion numbers in graphs, Czechoslovak Math. J. 3 (2008), 595–603. · Zbl 1174.05095
[2] W. Meng, On signed edge domination in digraphs, manuscript.
[3] S.M. Sheikholeslami, Signed total domination numbers of directed graphs, Util. Math.85 (2011), 273–279. · Zbl 1243.05098
[4] D.B. West, Introduction to Graph Theory, Prentice-Hall, Inc, 2000.
[5] B. Xu, L. Yinquan, On signed edge total domination numbers of graphs, J. Math. Practice Theory 39 (2009), 1–7. · Zbl 1212.05203
[6] B. Zelinka, On signed edge domination numbers of trees, Math. Bohem. 127 (2002), 49–55. · Zbl 0995.05112
[7] J. Zhao, B. Xu, On signed edge total domination numbers of graphs, J. Math. Res. Exposition 2 (2011), 209–214. · Zbl 1240.05237
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