# Hamiltonicities of Double Domination Critical and Stable Claw-Free Graphs

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## 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 uvE(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 uvE(G) and is said to be $k-γ×2-$-stable if γ×2 (Guv) = k for any uvE(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×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.

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# Discussiones Mathematicae Graph Theory

## The Journal of University of Zielona Góra

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Target Group

researchers in the fields of: colourings, partitions (general colourings), hereditary properties, independence and dominating structures (sets, paths, cycles, etc.), cycles, local properties, products of graphs

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