## Abstract

A graph *G* with the double domination number γ_{×2}(*G*) = *k* is said to be *k*- γ_{×2}-critical if γ_{×2} (*G* + *uv*) *< k* for any *uv* ∉ *E*(*G*). On the other hand, a graph *G* with γ_{×2} (*G*) = *k* is said to be _{×2} (*G* + *uv*) = *k* for any *uv* ∉ *E*(*G*) and is said to be _{×2} (*G*− *uv*) = *k* for any *uv* ∈ *E*(*G*). The problem of interest is to determine whether or not 2-connected *k*- γ_{×2}-critical graphs are Hamiltonian. In this paper, for *k* ≥ 4, we provide a 2-connected *k*- γ_{×2}-critical graph which is non-Hamiltonian. We prove that all 2-connected *k*-γ_{×2}-critical claw-free graphs are Hamiltonian when 2 ≤ *k* ≤ 5. We show that the condition claw-free when *k* = 4 is best possible. We further show that every 3-connected *k*- γ_{×2}-critical claw-free graph is Hamiltonian when 2 ≤ *k* ≤ 7. We also investigate Hamiltonian properties of