α-Domination perfect trees

Abstract

Let ∈ (0, 1) and let G = (V G , E G ) be a graph. According to Dunbar et al. [ -Domination, Discrete Math. 211 (2000) the minimum cardinality of an -dominating set of G and the -independent -domination number i (G) of G is the minimum cardinality of an -dominating set of G that is also -independent. A graph G is -domination perfect if (H ) = i (H ) for all induced subgraphs H of G.We characterize the -domination perfect trees in terms of their minimally forbidden induced subtrees. For ∈ (0, 1 2 ] there is exactly one such tree whereas for ∈ ( 1 2 , 1) there are infinitely many.

Upper Bounds for α-Domination Parameters

“…Notice that the concept of the α-rate domination number γ ×α (G) is 'opposite' to the α-independent α-domination number i α (G) as defined in [7]. It would be interesting to use a probabilistic method construction to obtain an upper bound for i α (G).…”

“…Interesting results on α-domination perfect graphs can be found in [7]. The problem of deciding whether γ α (G) ≤ k for a positive integer k is known to be N P -complete [9].…”