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Domination analysis
About: Domination analysis is a research topic. Over the lifetime, 3219 publications have been published within this topic receiving 35833 citations.
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TL;DR: It is proved that equality between these two parameters holds for trees and cactus graphs with no even cycles, and that associated decision problem for Roman { 2 } -domination is NP-complete, even for bipartite graphs.
123 citations
TL;DR: Upper bounds on the power domination number for a connected graph with at least three vertices and a connected claw-free cubic graph in terms of their order are presented.
120 citations
TL;DR: If G has order n with minimum degree ‐ and average degree d, then ∞£2(G) • ((ln(1 + d) + ln‐ + 1)=‐)n, where the minimum is taken over the n-dimensional cube C n.
Abstract: In a graph G, a vertex dominates itself and its neighbors. A subset S µ 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 ∞£2(G). A function f(p) is deflned, and it is shown that ∞£2(G) = minf(p), where the minimum is taken over the n-dimensional cube C n = fp = (p1;:::;pn) j pi 2 IR;0 • pi • 1;i = 1;:::;ng. Using this result, it is then shown that if G has order n with minimum degree ‐ and average degree d, then ∞£2(G) • ((ln(1 + d) + ln‐ + 1)=‐)n.
118 citations
TL;DR: The calculation of the domination number of all (n,m) grid graphs is concluded and Chang’s conjecture is proved saying that for every 16≤n≤m, γ(Gn, m) =⌊(n+2)(m+2)5⌋-4.5.
Abstract: In this paper, we conclude the calculation of the domination number of all (n,m) grid graphs Indeed, we prove Chang’s conjecture saying that for every 16≤n≤m, γ(Gn,m)=⌊(n+2)(m+2)5⌋-4
115 citations
TL;DR: It is shown that every 4-uniform hypergraph on n vertices and m edges has a transversal with no more than (5n + 4m)/21 vertices, and every graph with minimal degree at least 4 has total domination number at most 3n/7.
Abstract: The main result of this paper is that every 4-uniform hypergraph on n vertices and m edges has a transversal with no more than (5n + 4m)/21 vertices. In the particular case n = m, the transversal has at most 3n/7 vertices, and this bound is sharp in the complement of the Fano plane. Chvatal and McDiarmid [5] proved that every 3-uniform hypergraph with n vertices and edges has a transversal of size n/2. Two direct corollaries of these results are that every graph with minimal degree at least 3 has total domination number at most n/2 and every graph with minimal degree at least 4 has total domination number at most 3n/7. These two bounds are sharp.
115 citations