An upper bound on the 2-outer-independent domination number of a tree

Marcin Krzywkowski

doi:10.1016/j.crma.2011.10.005

Combinatorics

An upper bound on the 2-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. 21-22, pp. 1123-1125.

Abstract
Résumé

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

Un ensemble 2-dominant extérieurement-indépendant dʼun graphe G est un ensemble D de sommets de G tel que chaque sommet de $V (G) ∖ D$ a au moins deux voisins dans D, et lʼensemble $V (G) ∖ D$ est indépendant. Le nombre de 2-domination extérieurement-indépendante dʼun graphe G, noté par $γ_{2}^{oi} (G)$ , est le cardinal minimum dʼun ensemble 2-dominant extérieurement-indépendant de G. Nous prouvons lʼinégalité $γ_{2}^{oi} (T) ⩽ (n + l) / 2$ pour tout arbre non trivial T dʼordre n avec l feuilles, et nous caractérisons les arbres atteignant cette borne supérieure.

Received: 2010-12-09
Accepted: 2011-10-03
Published online: 2011-10-24

DOI: 10.1016/j.crma.2011.10.005

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. An upper bound on the 2-outer-independent domination number of a tree. Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1123-1125. doi : 10.1016/j.crma.2011.10.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.10.005/

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