Relating the annihilation number and the total domination number of a tree

Desormeaux, Wyatt J.; Haynes, Teresa W.; Henning, Michael A.

doi:10.1016/j.dam.2012.09.006

Cited by 18 publications

(17 citation statements)

References 8 publications

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“…To do this, we follow somewhat the same strategy as in Desormeaux et al (2013), where the total domination number of a tree and the annihilation number were considered.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

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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“…To do this, we follow somewhat the same strategy as in Desormeaux et al (2013), where the total domination number of a tree and the annihilation number were considered.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

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

Lyle

Patterson

2016

J Comb Optim

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“…Let γ t (G) denote the total domination number of a graph G. (For an extensive information on γ t see the book [13].) In [7,9] a parallel conjecture to Conjecture 1.1 was posed for the total domination number, that is, it was conjectured that γ t (G) ≤ a(G) + 1 (2) holds for every nontrivial connected graph G. This conjecture holds for graphs of minimum degree at least 3, and has been verified for trees [8] and for cactus graphs and block graphs [2]. The counterexamples to Conjecture 1.1 presented in this paper are far from being counterexamples for (2) since their total domination number is significantly smaller and, after all, the counterexamples to Conjecture 1.1 are cactus graphs for which (2) holds.…”

Section: Discussionmentioning

confidence: 99%

The annihilation number does not bound the 2-domination number from the above

Yue

Zhang

Zhu

et al. 2020

Discrete Mathematics

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The 2-domination number γ 2 (G) of a graph G is the minimum cardinality of a set S ⊆ V (G) such that every vertex from V (G) \ S is adjacent to at least two vertices in S. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of its edges. It was conjectured that γ 2 (G) ≤ a(G) + 1 holds for every connected graph G. The conjecture was earlier confirmed, in particular, for graphs of minimum degree 3, for trees, and for block graphs. In this paper, we disprove the conjecture by proving that the 2-domination number can be arbitrarily larger than the annihilation number. On the positive side we prove the conjectured bound for a large subclass of bipartite, connected cacti, thus generalizing a result of

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“…The relation between annihilation number and some graph parameters have been studied by several authors (see for example [1,6,12]).…”

Section: Introductionmentioning

confidence: 99%

Bounding the paired-domination number of a tree in terms of its annihilation number

Dehgardi

Sheikholeslami

Khodkar

2014

Filomat

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A paired-dominating set of a graph G = (V, E) with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G, denoted by γ pr (G), is the minimum cardinality of a paired-dominating set of G. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of edges in G. In this paper, we prove that for any tree T of order n ≥ 2, γ pr (T) ≤ 4a(T)+2 3 and we characterize the trees achieving this bound.

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Relating the annihilation number and the total domination number of a tree

Cited by 18 publications

References 8 publications

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

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

The annihilation number does not bound the 2-domination number from the above

Bounding the paired-domination number of a tree in terms of its annihilation number

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