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For vertices \( u \) and \( v \) in a connected graph \( G = (V, E) \), the monophonic detour distance \( d_m(u, v) \) is the length of a longest \( u-v \) monophonic path in \( G \). An \( u-v \) monophonic path of length \( d(u, v) \) is an \( u-v \) monophonic detour or an \( u-v \) \( m \)-detour. The set \( I_{d_m}[u, v] \) consists of all those vertices lying on an \( u-v \) \( m \)-detour in \( G \). Given a set \( S \) of vertices of \( G \), the union of all sets \( I_{d_m}[u, v] \) for \( u, v \in S \), is denoted by \( I_{d_m}[S] \). A set \( S \) is an \( m \)-detour convex set if \( I_{d_m}[S] = S \). The \( m \)-detour convex hull \( [S]_{d_m} \) of \( S \) in \( G \) is the smallest \( m \)-detour convex set containing \( S \).
A set \( S \) of vertices of \( G \) is an \( m \)-detour set if \( I_{d_m}[S] = V \) and the minimum cardinality of an \( m \)-detour set is the \( m \)-detour number \( md(G) \) of \( G \). A set \( S \) of vertices of \( G \) is an \( m \)-detour hull set if \( [S]_{d_m} = V \) and the minimum cardinality of an \( m \)-detour hull set is the \( m \)-detour hull number \( md_h(G) \) of \( G \).
Certain general properties of these concepts are studied. Bounds for the \( m \)-detour hull number of a graph are obtained. It is proved that every two integers \( a \) and \( b \) with \( 2 \leq a \leq b \) are realizable as the \( m \)-detour hull number and the \( m \)-detour number respectively, of some graph. Graphs \( G \) of order \( n \) for which \( md_h(G) = n \) or \( md_h(G) = n-1 \) are characterized. It is proved that for each triple \( a \), \( b \), and \( k \) of positive integers with \( a < b \) and \( k \geq 3 \), there exists a connected graph \( G \) with \( rad_m(G) = a \), \( diam_m(G) = b \), and \( md_h(G) = k \).