Combinatorics
Bounds on the vertex–edge domination number of a tree
Comptes Rendus. Mathématique, Volume 352 (2014) no. 5, pp. 363-366.

A vertex–edge dominating set of a graph G is a set D of vertices of G such that every edge of G is incident with a vertex of D or a vertex adjacent to a vertex of D. The vertex–edge domination number of a graph G, denoted by $γve(T)$, is the minimum cardinality of a vertex–edge dominating set of G. We prove that for every tree T of order $n⩾3$ with l leaves and s support vertices, we have $(n−l−s+3)/4⩽γve(T)⩽n/3$, and we characterize the trees attaining each of the bounds.

Un ensemble sommet–arête dominant d'un graphe G est un ensemble D de sommets de G tel que chaque arête de G soit incidente à un sommet de D ou à un sommet adjacent à un sommet de D. Le nombre de domination sommet–arête d'un graphe G, noté $γve(T)$, est le cardinal minimum d'un ensemble sommet–arête dominant de G. Nous prouvons que, pour chaque arbre T d'ordre $n⩾3$ avec l feuilles et des sommets s de soutien, que nous avons $(n−l−s+3)/4⩽γve(T)⩽n/3$, et nous caractérisons les arbres atteignant chacune des limites.

Accepted:
Published online:
DOI: 10.1016/j.crma.2014.03.017

Balakrishna Krishnakumari 1; Yanamandram B. Venkatakrishnan 1; Marcin Krzywkowski 2, 3

1 Department of Mathematics, SASTRA University, Tanjore, Tamil Nadu, India
2 Department of Mathematics, University of Johannesburg, South Africa
3 Faculty of Electronics, Telecommunications and Informatics, Gdansk University of Technology, Poland
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Balakrishna Krishnakumari; Yanamandram B. Venkatakrishnan; Marcin Krzywkowski. Bounds on the vertex–edge domination number of a tree. Comptes Rendus. Mathématique, Volume 352 (2014) no. 5, pp. 363-366. doi : 10.1016/j.crma.2014.03.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.03.017/

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