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For a connected graph \( G = (V, E) \) of order at least two, a chord of a path \( P \) is an edge joining two non-adjacent vertices of \( P \). A path \( P \) is called a monophonic path if it is a chordless path. A longest \( x-y \) monophonic path is called an \( x-y \) detour monophonic path. A set \( S \) of vertices of \( G \) is a monophonic set of \( G \) if each vertex \( v \) of \( G \) lies on an \( x-y \) monophonic path for some elements \( x \) and \( y \) in \( S \). The minimum cardinality of a monophonic set of \( G \) is the monophonic number of \( G \), denoted by \( m(G) \). A set \( S \) of vertices of \( G \) is a detour monophonic set of \( G \) if each vertex \( v \) of \( G \) lies on an \( x-y \) detour monophonic path for some \( x \) and \( y \) in \( S \). The minimum cardinality of a detour monophonic set of \( G \) is the detour monophonic number of \( G \) and is denoted by \( dm(G) \). We determine bounds for it and characterize graphs which realize these bounds. Also, for each pair \( a, b \) of integers with \( 2 \leq a \leq b \), we prove that there is a connected graph \( G \) with \( m(G) = a \) and \( dm(G) = b \).