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://archive.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://archive.numdam.org/articles/10.1051/ro/2018043/

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