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2-Edge connected dominating sets and 2-Connected dominating sets of a graph

Author

Listed:
  • Hengzhe Li

    (Henan Normal University)

  • Yuxing Yang

    (Henan Normal University)

  • Baoyindureng Wu

    (Xinjiang University)

Abstract

A $$k$$ k -connected (resp. $$k$$ k -edge connected) dominating set $$D$$ D of a connected graph $$G$$ G is a subset of $$V(G)$$ V ( G ) such that $$G[D]$$ G [ D ] is $$k$$ k -connected (resp. $$k$$ k -edge connected) and each $$v\in V(G)\backslash D$$ v ∈ V ( G ) \ D has at least one neighbor in $$D$$ D . The $$k$$ k -connected domination number (resp. $$k$$ k -edge connected domination number) of a graph $$G$$ G is the minimum size of a $$k$$ k -connected (resp. $$k$$ k -edge connected) dominating set of $$G$$ G , and denoted by $$\gamma _k(G)$$ γ k ( G ) (resp. $$\gamma '_k(G)$$ γ k ′ ( G ) ). In this paper, we investigate the relation of independence number and 2-connected (resp. 2-edge-connected) domination number, and prove that for a graph $$G$$ G , if it is $$2$$ 2 -edge connected, then $$\gamma '_2(G)\le 4\alpha (G)-1$$ γ 2 ′ ( G ) ≤ 4 α ( G ) - 1 , and it is $$2$$ 2 -connected, then $$\gamma _2(G)\le 6\alpha (G)-3$$ γ 2 ( G ) ≤ 6 α ( G ) - 3 , where $$\alpha (G)$$ α ( G ) is the independent number of $$G$$ G .

Suggested Citation

  • Hengzhe Li & Yuxing Yang & Baoyindureng Wu, 2016. "2-Edge connected dominating sets and 2-Connected dominating sets of a graph," Journal of Combinatorial Optimization, Springer, vol. 31(2), pages 713-724, February.
    • Handle: RePEc:spr:jcomop:v:31:y:2016:i:2:d:10.1007_s10878-014-9783-4
    DOI: 10.1007/s10878-014-9783-4
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    References listed on IDEAS

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    1. Ding-Zhu Du & Peng-Jun Wan, 2013. "Connected Dominating Set: Theory and Applications," Springer Optimization and Its Applications, Springer, edition 127, number 978-1-4614-5242-3, December.
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    Cited by:

    1. 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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