Every edge of the Markoff graph modulo p lies on a cycle, so connected Markoff graphs are 2-connected.
Non-planarity of Markoff graphs mod p.Commentarii Mathematici Helvetici, 99(1):111-148, 2024
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Sharp vertex connectivity of the Markoff graphs modulo $p$
Every edge of the Markoff graph modulo p lies on a cycle, so connected Markoff graphs are 2-connected.