Isolation Number versus Domination Number of Trees

Abstract

[Abstract] If 𝐺 = (Vɢ,Eɢ) is a graph of order n, we call 𝑆 ⊆ Vɢ an isolating set if the graph induced by Vɢ − Nɢ[𝑆] contains no edges. The minimum cardinality of an isolating set of 𝐺 is called the isolation number of 𝐺, and it is denoted by 𝜄(𝐺). It is known that 𝜄(𝐺) ≤ ⁿ⁄₃ and the bound is sharp. A subset 𝑆 ⊆ Vɢ is called dominating in 𝐺 if Nɢ[𝑆] = Vɢ. The minimum cardinality of a dominating set of 𝐺 is the domination number, and it is denoted by 𝛾(𝐺). In this paper, we analyze a family of trees 𝑇 where 𝜄(𝑇) = 𝛾(𝑇), and we prove that 𝜄(T) = ⁿ⁄₃ implies 𝜄(𝑇) = 𝛾(𝑇). Moreover, we give different equivalent characterizations of such graphs and we propose simple algorithms to build these trees from the connections of stars.