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

Lyle, Jeremy; Patterson, Sean C

doi:10.1007/s10878-016-0019-7

Cited by 7 publications

(3 citation statements)

References 5 publications

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“…We pose an open problem to characterize the quasi-trees G satisfying the equality γ t (G) = a(G) + 1. We add that a conjecture parallel to 1.1 has been posed also for the 2-domination number of a graph G. In [8] (see also [13]), the latter conjecture was verified for trees, and in [11] for block graphs. In [23], the conjecture was disproved by demonstrating that the 2-domination number can be arbitrarily larger than the annihilation number.…”

Section: Introductionmentioning

confidence: 71%

Relating the total domination number and the annihilation number for quasi-trees and some composite graphs

Hua¹,

Hua²,

Klavžar³

et al. 2021

Preprint

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The total domination number γ t (G) of a graph G is the cardinality of a smallest set D ⊆ V (G) such that each vertex of G has a neighbor in D. The annihilation number a(G) of G is the largest integer k such that there exist k different vertices in G with the degree sum at most m(G). It is conjectured that γ t (G) ≤ a(G) + 1 holds for every nontrivial connected graph G. The conjecture has been proved for graphs with minimum degree at least 3, trees, certain tree-like graphs, block graphs, and cactus graphs. In the main result of this paper it is proved that the conjecture holds for quasi-trees. The conjecture is verified also for some graph constructions including bijection graphs, Mycielskians, and the newly introduced universally-identifying graphs.

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Section: Introductionmentioning

confidence: 71%

Relating the total domination number and the annihilation number for quasi-trees and some composite graphs

Hua¹,

Hua²,

Klavžar³

et al. 2021

Preprint

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“…Hence, γ 2 (G) ≤ a(G) + 1 holds for any graph G with δ(G) ≥ 3. Desormeaux, Henning, Rall, and Yeo [9] followed with a confirmation of the conjecture for trees (see also [16] for another proof of it). Moreover, they have also characterized the trees that attain the equality in the conjecture.…”

Section: Introductionmentioning

confidence: 79%

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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“…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]. Namely, their result can be obtained if we replace the total domination number with the 2-domination number in Theorem 1.2.…”

Section: Theorem 12 ([5]mentioning

confidence: 94%

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

Cited by 7 publications

References 5 publications

Relating the total domination number and the annihilation number for quasi-trees and some composite graphs

Relating the total domination number and the annihilation number for quasi-trees and some composite graphs

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

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

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