In a graph a vertex is said to dominate itself and all its neighbours. A doubly dominating set of a graph G is a subset of vertices that dominates every vertex of G at least twice. A doubly dominating set is exact if every vertex of G is dominated exactly twice. We prove that the existence of an exact doubly dominating set is an NP-complete problem. We show that if an exact double dominating set exists then all such sets have the same size, and we establish bounds on this size. We give a constructive characterization of those trees that admit a doubly dominating set, and we establish a necessary and sufficient condition for the existence of an exact doubly dominating set in a connected cubic graph.
In a graph a vertex is said to dominate itself and all its neighbors. A double dominating set of a graph G is a subset of vertices that dominates every vertex of G at least twice. The double domination number of G, denoted $γ_{×2}(G)$, is the minimum cardinality among all double dominating sets of G. We consider the effects of vertex removal on the double domination number of a graph. A graph G is $γ_{×2}$-vertex critical graph ($γ_{×2}$-vertex stable graph, respectively) if the removal of any vertex different from a support vertex decreases (does not change, respectively) $γ_{×2}$(G). In this paper we investigate various properties of these graphs. Moreover, we characterize $γ_{×2}$-vertex critical trees and $γ_{×2}$-vertex stable trees.
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 $γ_{×2}(G) = min f(p)$, where the minimum is taken over the n-dimensional cube $Cⁿ = {p = (p₁,...,pₙ) | p_i ∈ IR, 0 ≤ p_i ≤ 1,i = 1,...,n}$. Using this result, it is then shown that if G has order n with minimum degree δ and average degree d, then $γ_{×2}(G) ≤ ((ln(1+d) + lnδ + 1)/δ)n$.
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 $γ_{×2}(G)$. If G ≠ C₅ is a connected graph of order n with minimum degree at least 2, then we show that $γ_{×2}(G) ≤ 3n/4$ and we characterize those graphs achieving equality.
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