Relating the annihilation number and the 2-domination number of a tree

Desormeaux, Wyatt J.; Henning, Michael A.; Rall, Douglas F.; Yeo, Anders

doi:10.1016/j.disc.2013.11.020

Cited by 15 publications

(14 citation statements)

References 7 publications

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“…A similar result was proved by Desormeaux, Henning, Rall, and Yeo [6] for the 2domination number of trees. Very recently, a different proof was given for the same statement by Lyle and Patterson [10].…”

Section: Theorem 12 ([5]supporting

confidence: 84%

Relating the total domination number and the annihilation number of cactus graphs and block graphs

Bujtás

Jakovac

2018

Ars Math. Contemp.

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The total domination number γ t (G) of a graph G is the order of a smallest set D ⊆ V (G) such that each vertex of G is adjacent to some vertex in D. The annihilation number a(G) of G is the largest integer k such that there exist k different vertices in G with degree sum of at most |E(G)|. It is conjectured that γ t (G) ≤ a(G) + 1 holds for every nontrivial connected graph G. The conjecture was proved for graphs with minimum degree at least 3, and remains unresolved for graphs with minimum degree 1 or 2. In this paper we establish the conjecture for cactus graphs and block graphs.

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Section: Theorem 12 ([5]supporting

confidence: 84%

Relating the total domination number and the annihilation number of cactus graphs and block graphs

Bujtás

Jakovac

2018

Ars Math. Contemp.

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“…Many known bounds on the domination number and the independence number depend only on the degree sequence, or on derived quantities such as the order, the size, the minimum degree, and the maximum degree . For a graph G with nonincreasing degree sequence

d = (d_{1}, \dots, d_{n})

, Slater observed

γ (G) \geq s ℓ (d)

where

s ℓ false(d false) = min \{0true k \in false[n false] : \sum_{i = 1}^{k} d_{i} \geq n - k\},

and Pepper observed

α (G) \leq a (d)

where

a false(d false) = max \{0true a \in false[n false] : \sum_{i = n - a + 1}^{n} d_{i} \leq \sum_{i = 1}^{n - a} d_{i}\} = n - min \{0true k \in false[n false] : \sum_{i = 1}^{k} d_{i} \geq \sum_{i = k + 1}^{n} d_{i}\}

is known as the annihilation number of G . Clearly,

γ_{trueprefixmin} false(d false) \geq s ℓ false(d false)

and

α_{trueprefixmax} false(d false) \leq a false(d false)

.…”

Section: Introductionmentioning

confidence: 99%

Smallest domination number and largest independence number of graphs and forests with given degree sequence

Gentner

Henning

Rautenbach

2017

Journal of Graph Theory

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“…This was subsequently proven by Desormeaux et al (2014) in the case when T is a tree. The bound considered for this paper is motivated by this result.…”

Section: Introductionmentioning

confidence: 74%

A note on the annihilation number and 2-domination number of a tree

Lyle

Patterson

2016

J Comb Optim

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, Desormeaux et al. (Discrete Math 319:15-23, 2014) proved a relationship between the annihilation number and 2-domination number of a tree. In this note, we provide a family of bounds for the 2-domination number of a tree based on the amount of vertices of small degree. This family of bounds extends current bounds on the 2-domination number of a tree, and provides an alternative proof for the relationship between the annihilation number and the 2-domination number of a tree that was shown by Desormeaux et al.

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Relating the annihilation number and the 2-domination number of a tree

Cited by 15 publications

References 7 publications

Relating the total domination number and the annihilation number of cactus graphs and block graphs

Relating the total domination number and the annihilation number of cactus graphs and block graphs

Smallest domination number and largest independence number of graphs and forests with given degree sequence

A note on the annihilation number and 2-domination number of a tree

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