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On the [1,2]-domination number of generalized Petersen graphs

Author

Listed:
  • Chen, Lily
  • Ma, Yingbin
  • Shi, Yongtang
  • Zhao, Yan

Abstract

A dominating set in a graph G=(V,E) is a subset S of V such that N[S]=V, that is, each vertex of G either belongs to S or is adjacent to at least one vertex in S. The minimum cardinality of a dominating set in G is called the domination number, denoted by γ(G). A subset S of V is a [1,2]-set if, for every vertex v ∈ V∖S, v is adjacent to at least one but no more than two vertices in S. The [1,2]-domination number of a graph G, denoted by γ[1, 2](G), is the minimum cardinality of a [1, 2]-set of Chellali et al. gave some bounds for γ[1, 2](G) and proposed the following problem: which graphs satisfy γ(G)=γ[1,2](G). Ebrahimi et al. determined the exact value of the domination number for generalized Petersen graphs P(n, k) when k ∈ {1, 2, 3}. In this paper, we determine the exact values of γ[1, 2](P(n, k)) for k ∈ {1, 2, 3}. We also show that γ[1,2](P(n,k))=γ(P(n,k)) for k=1 and k=3, respectively, while for k=2,γ[1, 2](P(n, k)) ≠ γ(P(n, k)) except for n=6,7,9,12.

Suggested Citation

  • Chen, Lily & Ma, Yingbin & Shi, Yongtang & Zhao, Yan, 2018. "On the [1,2]-domination number of generalized Petersen graphs," Applied Mathematics and Computation, Elsevier, vol. 327(C), pages 1-7.
  • Handle: RePEc:eee:apmaco:v:327:y:2018:i:c:p:1-7
    DOI: 10.1016/j.amc.2018.01.013
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    Cited by:

    1. Gao, Zhipeng & Lei, Hui & Wang, Kui, 2020. "Rainbow domination numbers of generalized Petersen graphs," Applied Mathematics and Computation, Elsevier, vol. 382(C).
    2. Ma, Yuede & Cai, Qingqiong & Yao, Shunyu, 2019. "Integer linear programming models for the weighted total domination problem," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 146-150.
    3. Yue, Jun & Wei, Meiqin & Li, Min & Liu, Guodong, 2018. "On the double Roman domination of graphs," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 669-675.

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