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Here are some comments that may be useful for others in a similar position to me when I asked this question a few months ago: I'll give an explicit $MSO_2$ formula for simple cycles. This is adapted from the $MSO_2$ formula for Hamiltonian cycles from section $7.4.1$ of Parametrized Algorithms by Cygan et. al. Let $G = (V,E)$ be a graph. For $X \subseteq E$...


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