α-Domination perfect trees

AbstractLet 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|⩾αdG(u) for all u∈VG⧹D. Similarly, we define a set D⊆VG to be an α-independent set of G if |NG(u)∩D|⩽αdG(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∈(0,12] there is exactly one such tree whereas for α∈(12,1) there are infinitely many