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

Hengzhe Li, Yuxing Yang and Baoyindureng Wu ()
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Hengzhe Li: Henan Normal University
Yuxing Yang: Henan Normal University
Baoyindureng Wu: Xinjiang University

Journal of Combinatorial Optimization, 2016, vol. 31, issue 2, No 15, 713-724

Abstract: 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 .

Keywords: Connected dominating set; Dominating set; Independent set (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10878-014-9783-4

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