A lower bound on the total outer-independent domination number of a tree

Marcin Krzywkowski

doi:10.1016/j.crma.2010.11.021

Combinatorics

A lower bound on the total outer-independent domination number of a tree

Presented by: the Editorial Board

Marcin Krzywkowski ¹

¹ Faculty of Electronics, Telecommunications and Informatics, Gdańsk University of Technology, Narutowicza 11/12, 80-233 Gdańsk, Poland

Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 7-9.

Abstract
Résumé

A total outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbour in D, and the set $V (G) ∖ D$ is independent. The total outer-independent domination number of a graph G, denoted by $γ_{t}^{o i} (G)$ , is the minimum cardinality of the total outer-independent dominating set of G. We prove that for every nontrivial tree T of order n with l leaves we have $γ_{t}^{o i} (T) ⩾ (2 n - 2 l + 2) / 3$ , and we characterize the trees attaining this lower bound.

Un sous-ensemble totalement dominant et extérieurement indépendant d'un graphe est un sous-ensemble D des sommets de G tel que chaque sommet de G ait un voisin dans D et l'ensemble $V (G) ∖ D$ soit indépendant. Le plus petit cardinal d'un tel sous-ensemble est noté $γ_{t}^{o i} (G)$ . Nous démontrons que pour tout arbre T non trivial, d'ordre n avec l feuilles, nous avons $γ_{t}^{o i} (T) ⩾ (2 n - 2 l + 2) / 3$ . De plus, nous caractérisons les arbres réalisant cette borne inférieure.

Received: 2010-02-28
Accepted: 2010-11-19
Published online: 2010-12-18

DOI: 10.1016/j.crma.2010.11.021

Author's affiliations:

Marcin Krzywkowski ¹

¹ Faculty of Electronics, Telecommunications and Informatics, Gdańsk University of Technology, Narutowicza 11/12, 80-233 Gdańsk, Poland

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Marcin Krzywkowski. A lower bound on the total outer-independent domination number of a tree. Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 7-9. doi : 10.1016/j.crma.2010.11.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.11.021/

References
Cited by

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

[2] E. Cockayne; R. Dawes; S. Hedetniemi Total domination in graphs, Networks, Volume 10 (1980), pp. 211-219

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

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

[5] M. Krzywkowski, Total outer-independent domination in graphs, submitted for publication.

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