Maximum sizes of graphs with given domination parameters

“…However, if we assume that G is connected, and forbid a constant number of counterexamples, then the first (see [7]) and third (see [10]) constants can be improved from , respectively. In [4], an upper bound was given on the number of edges in a graph with a given order and given total domination number. The bounds in [4] are sharp, but also quadratic in nature.…”

Section: Theorem 11 If G Is a Graph With Minimum Degree δ(G) Then mentioning

confidence: 99%

“…In [4], an upper bound was given on the number of edges in a graph with a given order and given total domination number. The bounds in [4] are sharp, but also quadratic in nature. Henning [6] gave the following linear bound involving the order, total domination number and maximum degree.…”

Section: Theorem 11 If G Is a Graph With Minimum Degree δ(G) Then mentioning

confidence: 99%

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Relationships between total domination, order, size, and maximum degree of graphs

Yeo

2007

Journal of Graph Theory

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A total dominating set, S, in a graph, G, has the property that every vertex in G is adjacent to a vertex in S. The total dominating number, γt(G) of a graph G is the size of a minimum total dominating set in G. Let G be a graph with no component of size one or two and with Δ(G) ≥ 3. In 6, it was shown that |E(G)| ≤ Δ(G) (|V(G)|–γt(G)) and conjectured that |E(G)| ≤ (Δ(G)+3) (|V(G)|–γt(G))/2 holds. In this article, we prove that $\leq (\Delta(G)+ 2\sqrt{\Delta}) (|V(G)| - \gamma_{t}(G))/2$ holds and that the above conjecture is false as there for every Δ exist Δ‐regular bipartite graphs G with |E(G)| ≥ (Δ+0.1 ln(Δ)) (|V(G)|–γt(G))/2. The last result also disproves a conjecture on domination numbers from 8. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 325–337, 2007

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“…Dankelmann et al [4] proved a Vizing-like relation between the size and the total domination number of a graph of given order. Sanchis [15] showed that if we restrict our attention to connected graphs with total domination number at least 5, then the bound in [4] can be improved slightly.…”

Section: Introductionmentioning

confidence: 99%

Equality in a linear Vizing-like relation that relates the size and total domination number of a graph

Henning¹,

Joubert²

2013

Discrete Applied Mathematics

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Let G be a graph each component of which has order at least 3, and let G have order n, size m, total domination number γt and maximum degree ∆(G). Let ∆ = 3 if ∆(G) = 2 and ∆ = ∆(G) if ∆(G) ≥ 3. It is known [J. Graph Theory 49 (2005), 285-290; J. Graph Theory 54 (2007), 350-353] that m ≤ ∆(n − γt). In this paper we characterize the extremal graphs G satisfying m = ∆(n − γt).

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Maximum sizes of graphs with given domination parameters

Cited by 33 publications

References 6 publications

A linear Vizing‐like relation relating the size and total domination number of a graph

A linear Vizing‐like relation relating the size and total domination number of a graph

Relationships between total domination, order, size, and maximum degree of graphs

Equality in a linear Vizing-like relation that relates the size and total domination number of a graph

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