2016
DOI: 10.1007/s10878-016-0019-7
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A note on the annihilation number and 2-domination number of a tree

Abstract: , 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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Cited by 7 publications
(3 citation statements)
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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%
“…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%
“…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%
“…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%