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For two vertices \( u \) and \( v \) in a connected graph \( G \), the detour distance \( D(u,v) \) from \( u \) to \( v \) is defined as the length of a longest \( u-v \) path in \( G \). The detour eccentricity \( e_D(v) \) of a vertex \( v \) in \( G \) is the maximum detour distance from \( v \) to a vertex of \( G \). The detour radius \( \text{rad}_D(G) \) of \( G \) is the minimum detour eccentricity among the vertices of \( G \), while the detour diameter \( \text{diam}_D(G) \) of \( G \) is the maximum detour eccentricity among the vertices of \( G \). It is shown that
\[\text{rad}_D(G) < \text{diam}_D(G) < 2\text{rad}_D(G)\] for every connected graph \( G \) and that every pair \( a,b \) of positive integers with \( a \leq b \leq 2a \) is realizable as the detour radius and detour diameter of some connected graph. The detour center of \( G \) is the subgraph induced by those vertices of \( G \) having detour eccentricity \( \text{rad}_D(G) \). A connected graph \( G \) is detour self-centered if \( G \) is its own detour center. The detour periphery of \( G \) is the subgraph induced by the vertices of \( G \) having detour eccentricity \( \text{diam}_D(G) \). It is shown that every graph is the detour center of some connected graph. Detour self-centered graphs are investigated. We present sufficient conditions for a graph to be the detour periphery of some connected graph. Several classes of graphs that are not the detour periphery of any connected graph are determined.