P. Kaemawichanurat
Hamiltonicities of double domination critical and stable claw-free graphs
Discussiones Mathematicae Graph Theory
Received 30.10.2017, Revised 07.05.2018, Accepted 07.05.2018, doi: 10.7151/dmgt.2148

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 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.
double domination, critical, stable, Hamiltonian