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A path \(x_1, x_2, \dots, x_n\) in a connected graph \( G \) that has no edge \( x_i x_j \) \((j \geq i+3)\) is called a \textit{monophonic-triangular path} or \textit{mt}-path. A non-empty subset \( M \) of \( V(G) \) is a \textit{monophonic-triangular set} or \textit{mt-set} of \( G \) if every member in \( V(G) \) exists in a \textit{mt}-path joining some pair of members in \( M \). The \textit{monophonic-triangular number} or \textit{mt-number} is the lowest cardinality of an \textit{mt}-set of \( G \) and it is symbolized by \( mt(G) \). The general properties satisfied by \textit{mt}-sets are discussed. Also, we establish \( mt \)-number boundaries and discover similar results for a few common graphs. Graphs \( G \) of order \( p \) with \( mt(G) = p \), \( p – 1 \), or \( p – 2 \) are characterized.