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A simple acyclic graphoidal cover of a graph \( G \) is a collection \( \psi \) of paths in \( G \) such that every path in \( \psi \) has at least two vertices, every vertex of \( G \) is an internal vertex of at most one path in \( \psi \), every edge of \( G \) is in exactly one path in \( \psi \), and any two paths in \( \psi \) have at most one vertex in common. The minimum cardinality of a simple acyclic graphoidal cover of \( G \) is called the simple acyclic graphoidal covering number of \( G \) and is denoted by \( \eta_{as}(G) \). A simple acyclic graphoidal cover \( \psi \) of \( G \) with \( |\psi| = \eta_{as}(G) \) is called a minimum simple acyclic graphoidal cover of \( G \). Two minimum simple acyclic graphoidal covers \( \psi_1 \) and \( \psi_2 \) of \( G \) are said to be isomorphic if there exists an automorphism \( \alpha \) of \( G \) such that \( \psi = \{\alpha(P) : P \in \psi_1\} \). In this paper, we characterize trees, unicyclic graphs, and wheels in which any two minimum simple acyclic graphoidal covers are isomorphic.