Comptes Rendus
no. 5
Combinatorics
Bounds on the vertex–edge domination number of a tree
[Bornes sur le nombre de domination sommet–arête d'un arbre]

Présenté par : the Editorial Board

Balakrishna Krishnakumari1 ; Yanamandram B. Venkatakrishnan1 ; Marcin Krzywkowski2, 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
Comptes Rendus. Mathématique, Volume 352 (2014) no. 5, pp. 363-366.

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.

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.

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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/

[1] M. Chellali; T. Haynes A note on the total domination number of a tree, J. Comb. Math. Comb. Comput., Volume 58 (2006), pp. 189-193

[2] T. Haynes; S. Hedetniemi; P. Slater Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998

[3] Domination in Graphs: Advanced Topics (T. Haynes; S. Hedetniemi; P. Slater, eds.), Marcel Dekker, New York, 1998

[4] M. Krzywkowski A lower bound on the total outer-independent domination number of a tree, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011), pp. 7-9

[5] M. Lemańska Lower bound on the domination number of a tree, Discuss. Math., Graph Theory, Volume 24 (2004), pp. 165-169

[6] J. Lewis; S. Hedetniemi; T. Haynes; G. Fricke Vertex–edge domination, Util. Math., Volume 81 (2010), pp. 193-213

[7] J. Peters Theoretical and algorithmic results on domination and connectivity, Clemson University, 1986 (PhD thesis)

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