Bounds for signed double Roman k-domination in trees
RAIRO - Operations Research - Recherche Opérationnelle, Tome 53 (2019) no. 2, pp. 627-643.

Let k ≥ 1 be an integer and G be a simple and finite graph with vertex set V ( G ) . A signed double Roman k -dominating function (SDR k DF) on a graph G is a function f : V ( G ) → { − 1 , 1 , 2 , 3 } such that (i) every vertex v with f ( v ) = − 1 is adjacent to at least two vertices assigned a 2 or to at least one vertex w with f ( w ) = 3 , (ii) every vertex v with f ( v ) = 1 is adjacent to at least one vertex w with f ( w ) ≥ 2 and (iii) ∑ u ∈ N [ v ] f ( u ) ≥ k holds for any vertex v . The weight of a SDR k DF f is ∑ u ∈ V ( G ) f ( u ) X holds for any vertex $v$. The weight of a SDRkDF f is ∑ f(u), and the minimum weight of a SDRkDF is the signed double Roman k-domination number . The weight of a SDRkDF f is ∑ γ s d R k ( G ) of G . In this paper, we investigate the signed double Roman k -domination number γ s d R k ( T ) for 2 ≤ k ≤ 6 and classify all extremal trees.

DOI : 10.1051/ro/2018043
Classification : 05C69
Yang, Hong 1 ; Wu, Pu 1 ; Nazari-Moghaddam, Sakineh 1 ; Sheikholeslami, Seyed Mahmoud 1 ; Zhang, Xiaosong 1 ; Shao, Zehui 1 ; Tang, Yuan Yan 1

1 
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     author = {Yang, Hong and Wu, Pu and Nazari-Moghaddam, Sakineh and Sheikholeslami, Seyed Mahmoud and Zhang, Xiaosong and Shao, Zehui and Tang, Yuan Yan},
     title = {Bounds for signed double {Roman} k-domination in trees},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {627--643},
     publisher = {EDP-Sciences},
     volume = {53},
     number = {2},
     year = {2019},
     doi = {10.1051/ro/2018043},
     mrnumber = {3959475},
     zbl = {1440.05164},
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     url = {http://www.numdam.org/articles/10.1051/ro/2018043/}
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Yang, Hong; Wu, Pu; Nazari-Moghaddam, Sakineh; Sheikholeslami, Seyed Mahmoud; Zhang, Xiaosong; Shao, Zehui; Tang, Yuan Yan. Bounds for signed double Roman k-domination in trees. RAIRO - Operations Research - Recherche Opérationnelle, Tome 53 (2019) no. 2, pp. 627-643. doi : 10.1051/ro/2018043. http://www.numdam.org/articles/10.1051/ro/2018043/

[1] H. Abdollahzadeh Ahangar, J. Amjadi, M. Atapour, M. Chellali and S.M. Sheikholeslami, Double roman trees. To appear in: Ars Combin. (2019). | MR | Zbl

[2] H. Abdollahzadeh Ahangar, J. Amjadi, M. Chellali, S. Nazari-Moghaddam and S.M. Sheikholeslami, Trees with double Roman domination number twice the domination number plus two. To appear in: Iran. J. Sci. Technol. Trans. A Sci. (2019). | MR

[3] H. Abdollahzadeh Ahangar, M. Chellali and S.M. Sheikholeslami, On the double Roman domination in graphs. Discrete Appl. Math. 232 (2017) 1–7. | MR | Zbl

[4] H. Abdollahzadeh Ahangar, M. Chellali and S.M. Sheikholeslami, Signed double Roman domination in graphs. Discrete Appl. Math. 257 (2019) 1–11. | MR | Zbl

[5] H. Abdollahzadeh Ahangar, M.A. Henning, C. Löwenstein, Y. Zhao and V. Samodivkin, Signed Roman domination in graphs. J. Comb. Optim. 27 (2014) 241–255. | MR | Zbl

[6] J. Amjadi, S. Nazari-Moghaddam, S.M. Sheikholeslami and L. Volkmann, An upper bound on the double Roman domination number. J. Comb. Optim. 36 (2018). Doi: 10.1007/s10878-018-0286-6. | MR | Zbl

[7] R.A. Beeler, T.W. Haynes and S.T. Hedetniemi, Double Roman domination. Discrete Appl. Math. 211 (2016) 23–29. | MR | Zbl

[8] O. Favaron, Signed domination in regular graphs. Discrete Math. 158 (1996) 287–293. | MR | Zbl

[9] M.A. Henning and L. Volkmann, Signed Roman k-domination in graphs. Graphs Combin. 32 (2016) 175–190. | MR | Zbl

[10] M.A. Henning and L. Volkmann, Signed Roman k-domination in trees. Discrete Appl. Math. 186 (2015) 98–105. | MR | Zbl

[11] A. Mahmoodi, On the signed Roman edge k-domination in graphs. Commun. Comb. Optim. 2 (2017) 57–64. | MR | Zbl

[12] Z. Shao, S. Klavzar, Z. Li, P. Wu and J. Xu, On the signed Roman k-domination: complexity and thin torus graphs. Discrete Appl. Math. 233 (2017) 175–186. | MR | Zbl

[13] S.M. Sheikholeslami and L. Volkmann, The signed Roman domatic number of a digraph. Electron. J. Graph Theory Appl. 3 (2015) 85–93. | MR | Zbl

[14] S.M. Sheikholeslami and L. Volkmann, Signed Roman domination number in digraphs. J. Comb. Optim. 30 (2015) 456–467. | MR | Zbl

[15] L. Volkmann, The double Roman domatic number of a graph. J. Combin. Math. Combin. Comput. 104 (2018) 205–215. | MR | Zbl

[16] L. Volkmann, Double Roman domination and domatic numbers of graphs. Commun. Comb. Optim. 3 (2018) 71–77. | MR | Zbl

[17] L. Volkmann, The signed Roman k-domatic number of a graph. Discrete Appl. Math. 180 (2015) 150–157. | MR | Zbl

[18] L. Volkmann, Signed Roman k-domination in digraphs. Graphs Combin. 32 (2016) 1217–1227. | MR | Zbl

[19] L. Volkmann, The signed Roman k-domatic number of digraphs. Aust. J. Combin. 64 (2016) 444–457. | MR | Zbl

[20] X. Zhang, Z. Li, H. Jiang and Z. Shao, Double Roman domination in trees. Inform. Process. Lett. 134 (2018) 31–34. | MR | Zbl

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