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A set \( D \) of vertices in a graph \( G \) is irredundant if every vertex \( v \) in \( D \) has at least one private neighbour in \( N[v, G] \setminus N[D \setminus \{v\}, G] \). A set \( D \) of vertices in a graph \( G \) is a minimal dominating set of \( G \) if \( D \) is irredundant and every vertex in \( V(G) \setminus D \) has at least one neighbour in \( D \). Further, irredundant sets and minimal dominating sets of maximal cardinality are called \( IR \)-sets and \( \Gamma \)-sets, respectively. A set \( I \) of the vertex set of a graph \( G \) is independent if no two vertices in \( I \) are adjacent, and independent sets of maximal cardinality are called \( \alpha \)-sets.
In this paper, we prove that bipartite graphs and chordal graphs have a unique \( \alpha \)-set if and only if they have a unique \( \Gamma \)-set if and only if they have a unique \( IR \)-set. Some related results are also presented.