5 research outputs found

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    On double domination in graphs

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    In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number γ ×2(G). A function f(p) is defined, and it is shown that γ ×2(G) = minf(p), where the minimum is taken over the n-dimensional cube Cn = {p = (p1,…,pn) | pi ∈ IR, 0 ≤ pi ≤ 1,i = 1,…,n}. Using this result, it is then shown that if G has order n with minimum degree δ and average degree d, then γ×2(G) ≤ ((ln(1+d)+lnδ+1)/δ)n

    Trees whose 2-domination subdivision number is 2

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    A set SS of vertices in a graph G=(V,E)G = (V,E) is a 22-dominating set if every vertex of V∖SV\setminus S is adjacent to at least two vertices of SS. The 22-domination number of a graph GG, denoted by γ2(G)\gamma_2(G), is the minimum size of a 22-dominating set of GG. The 22-domination subdivision number sdγ2(G)sd_{\gamma_2}(G) is the minimum number of edges that must be subdivided (each edge in GG can be subdivided at most once) in order to increase the 22-domination number. The authors have recently proved that for any tree TT of order at least 33, 1≤sdγ2(T)≤21 \leq sd_{\gamma_2}(T)\leq 2. In this paper we provide a constructive characterization of the trees whose 22-domination subdivision number is 22

    Characterizations of Trees With Equal Paired and Double Domination Numbers

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    A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, and a double dominating set is a dominating set that dominates every vertex of G at least twice. We show that for trees, the paired-domination number is less than or equal to the double domination number, solving a conjecture of Chellali and Haynes. Then we characterize the trees having equal paired and double domination numbers
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