We study the stability of some classes of graphs defined in terms of forbidden subgraphs under the closure operation introduced by the second author. Using these results, we prove that every 2-connected claw-free and P_7-free, or claw-free and Z_4-free, or claw-free and eiffel-free graph is either hamiltonian or belongs to a certain class of exceptions (all of them having connectivity 2).
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