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

Self Cite

“…The second case, Case 2, is when n 1 + n 2 = 0. This case is not covered in the proof of Theorem 3 in [1]. Our aim is to present a proof of this case.…”

mentioning

confidence: 84%

mentioning

confidence: 98%

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Erratum to: “A linear vizing‐like relation relating the size and total domination number of a graph”

Shan

Kang

Henning

2007

Self Cite

“…(Henning's Lemma [7]) If G is a graph with n vertices and m edges, where is an integer, γ t (B) ≤ (p − 3)/2 in this case, and adding v to a smallest total dominating set of B − T yields the desired set.…”

Section: Balloons and Total Dominationmentioning

confidence: 99%

Balloons, cut‐edges, matchings, and total domination in regular graphs of odd degree

Suil¹,

West²

2009

A balloon in a graph G is a maximal 2-edge-connected subgraph incident to exactly one cut-edge of G. Let b(G) be the number of balloons, let c(G) be the number of cut-edges, and let α ′ (G) be the maximum size of a matching. Let F n,r be the family of connected (2r + 1)-regular graphs with n vertices, and let b = max{b(G) : G ∈ F n,r }. For G ∈ F n,r , we prove the sharp inequalities c(G) ≤ r(n−2)−2 2r 2 +2r−14r 2 +4r−2 , we obtain a simple proof of the bound α ′ (G) ≥ proved by Henning and Yeo. For each of these bounds and each r, the approach using balloons allows us to determine the infinite family where equality holds. For the total domination number γ t (G) of a cubic graph, we prove γ t (G) ≤ 3 for cubic graphs, this improves the known inequality γ t (G) ≤ α ′ (G).

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

Yeo

2007

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