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- Title
Signed Roman $$k$$ -Domination in Graphs.
- Authors
Henning, Michael; Volkmann, Lutz
- Abstract
Let $$k\ge 1$$ be an integer, and let $$G$$ be a finite and simple graph with vertex set $$V(G)$$ . A signed Roman $$k$$ -dominating function (SRkDF) on a graph $$G$$ is a function $$f:V(G)\rightarrow \{-1,1,2\}$$ satisfying the conditions that (i) $$\sum _{x\in N[v]}f(x)\ge k$$ for each vertex $$v\in V(G)$$ , where $$N[v]$$ is the closed neighborhood of $$v$$ , and (ii) every vertex $$u$$ for which $$f(u)=-1$$ is adjacent to at least one vertex $$v$$ for which $$f(v)=2$$ . The weight of an SRkDF $$f$$ is $$w(f)=\sum _{v\in V(G)}f(v)$$ . The signed Roman $$k$$ -domination number $$\gamma _{sR}^k(G)$$ of $$G$$ is the minimum weight of an SRkDF on $$G$$ . In this paper we initiate the study of the signed Roman $$k$$ -domination number of graphs, and we present different bounds on $$\gamma _{sR}^k(G)$$ . In addition, we determine the signed Roman $$k$$ -domination number of some classes of graphs. Some of our results are extensions of well-known properties of the signed Roman domination number $$\gamma _{sR}(G)=\gamma _{sR}^1(G)$$ , introduced and investigated by Ahangar et al. (J Comb Optim doi:, ).
- Subjects
DOMINATING set; GRAPH theory; INTEGERS; GEOMETRIC vertices; SET theory; MATHEMATICAL bounds
- Publication
Graphs & Combinatorics, 2016, Vol 32, Issue 1, p175
- ISSN
0911-0119
- Publication type
Article
- DOI
10.1007/s00373-015-1536-3