Strong equality of domination parameters in graphs

Abstract

Let Q ₁ and Q ₂ be properties of vertex subsets of a graph, and assume that every subset of V(G) with property Q ₂ also has property Q ₁. Let ψ ₁ (G) and ψ ₂ (G), respectively, denote the minimum cardinalities of sets with properties Q ₁ and Q ₂, respectively. Then ψ ₁ (G) iψ ₂(G). If ψ ₁ (G) = ψ ₂ (G) and every ψ ₁(G)-set is also ψ ₂ (G)-set, then we say ψ ₁(G) strongly equals ψ ₂(G), written ψ ₁(G) ≡ iψ ₂(G). In this paper, we characterize the trees and cubic graphs with γ(G) ≡ γ _C(G) and γ _t (G) ≡ γ _c(G), where γ(G), γ _t(G) and γ _C(G) are the domination, total domination and connected domination numbers, respectively.