Hamiltonicity of double domination critical claw-free graphs

Pawaton Kaemawichanurat
2018 Discussiones Mathematicae Graph Theory  
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 k-γ + ×2 -stable if γ ×2 (G + uv) = k for any uv / ∈ E(G) and is said to be k-γ − ×2 -stable if γ ×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
more » ... 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 k-γ + ×2 -stable graphs and k-γ − ×2 -stable graphs.
doi:10.7151/dmgt.2148 fatcat:zcqolhixk5ccdnvhhtg3zrp5dm