α-Domination perfect trees

Abstract

Let α ∈ (0, 1) and let G = (VG, EG) be a graph. According to Dunbar et al. [α-Domination, Discrete Math. 211 (2000) 11-26], a set D ⊆ VG is an α-dominating set of G if | NG (u) ∩ D | ≥ α dG (u) for all u ∈ VG {minus 45 degree rule} D. Similarly, we define a set D ⊆ VG to be an α-independent set of G if | NG (u) ∩ D | ≤ α dG (u) for all u ∈ D. The α-domination number γα (G) of G is the minimum cardinality of an α-dominating set of G and the α-independent α-domination number iα (G) of G is the minimum cardinality of an α-dominating set of G that is also α-independent. A graph G is α-domination perfect if γα (H) = iα (H) for all induced subgraphs H of G. We characterize the α-domination perfect trees in terms of their minimally forbidden induced subtrees. For α ∈ (0, frac(1, 2)] there is exactly one such tree whereas for α ∈ (frac(1, 2), 1) there are infinitely many. © 2007 Elsevier B.V. All rights reserved.