Combinatorics
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.

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 $γtoi(G)$, 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 $γtoi(T)⩾(2n−2l+2)/3$, 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 $γtoi(G)$. Nous démontrons que pour tout arbre T non trivial, d'ordre n avec l feuilles, nous avons $γtoi(T)⩾(2n−2l+2)/3$. De plus, nous caractérisons les arbres réalisant cette borne inférieure.

Accepted:
Published online:
DOI: 10.1016/j.crma.2010.11.021

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/

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

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