Geodesic
Bounds on the Signed Roman k-Domination Number of a Digraph
Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 1, pp. 67-79.

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Let k be a positive integer. A signed Roman k-dominating function (SRkDF) on a digraph D is a function f : V (D) →{−1, 1, 2 } satisfying the conditions that (i) Σ_ x ∈ N^− [v] f(x) ≥ k for each v ∈ V (D), where N^− [v] is the closed in-neighborhood of v, and (ii) each vertex u for which f(u) = −1 has an in-neighbor v for which f(v) = 2. The weight of an SRkDF f is Σ_ v ∈ V (D) f(v). The signed Roman k-domination number γ_sR^k (D) of a digraph D is the minimum weight of an SRkDF on D. We determine the exact values of the signed Roman k-domination number of some special classes of digraphs and establish some bounds on the signed Roman k-domination number of general digraphs. In particular, for an oriented tree T of order n, we show that γ_sR^2 (T) ≥ (n + 3)//2, and we characterize the oriented trees achieving this lower bound.
Mots-clés : signed Roman k-dominating function, signed Roman k-domination number, digraph, oriented tree 
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Chen, Xiaodan; Hao, Guoliang; Volkmann, Lutz. Bounds on the Signed Roman k-Domination Number of a Digraph. Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 1, pp. 67-79. https://geodesic-test.mathdoc.fr/item/DMGT_2019_39_1_a6/

[1] H.A. Ahangar, M.A. Henning, C. Löwenstein, Y. Zhao and V. Samodivkin, Signed Roman domination in graphs, J. Comb. Optim. 27 (2014) 241-255. doi: 10.1007/s10878-012-9500-0

[2] Y. Caro and M.A. Henning, Directed domination in oriented graphs, Discrete Appl. Math. 160 (2012) 1053-1063. doi: 10.1016/j.dam.2011.12.027

[3] G. Chartrand and L. Lesniak, Graphs and Digraphs, Fourth Ed. (Chapman and Hall, Boca Raton, FL, 2005).

[4] J. Cyman, Total outer-connected domination in trees, Discuss. Math. Graph Theory 3 (2010) 377-383. doi: /10.7151/dmgt.1500

[5] G. Hao and J. Qian, On the sum of out-domination number and in-domination number of digraphs, Ars Combin. 119 (2015) 331-337.

[6] F. Harary, R.Z. Norman and D. Cartwright, Structural Models (Wiley, New York, 1965).

[7] M.A. Henning and V. Naicker, Bounds on the disjunctive total domination number of a tree, Discuss. Math. Graph Theory 36 (2016) 153-171. doi: 10.7151/dmgt.1854

[8] M.A. Henning and L. Volkmann, Signed Roman k-domination in trees, Discrete Appl. Math. 186 (2015) 98-105. doi: 10.1016/j.dam.2015.01.019

[9] M.A. Henning and L. Volkmann, Signed Roman k-domination in graphs, Graphs Combin. 32 (2016) 175-190. doi: 10.1007/s00373-015-1536-3

[10] S.M. Sheikholeslami and L. Volkmann, Signed Roman domination in digraphs, J. Comb. Optim. 30 (2015) 456-467. doi: 10.1007/s10878-013-9648-2

[11] L. Volkmann, Signed Roman k-domination in digraphs, Graphs Combin. 32 (2016) 1217-1227. doi: 10.1007/s00373-015-1641-3

[12] B. Zelinka, Signed domination numbers of directed graphs, Czechoslovak Math. J. 55 (2005) 479-482. doi: 10.1007/s10587-005-0038-5.