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For two vertices \(u\) and \(v\) in a graph \(G = (V, E)\), the \emph{detour distance} \(D(u,v)\) is the length of a longest \(u\)-\(v\) path in \(G\). A \(u\)-\(v\) path of length \(D(u, v)\) is called a \(u\)-\(v\) \emph{detour}. A set \(S \subseteq V\) is called an \emph{edge detour set} if every edge in \(G\) lies on a detour joining a pair of vertices of \(S\). The \emph{edge detour number} \(dn_1(G)\) of \(G\) is the minimum order of its edge detour sets, and any edge detour set of order \(dn_1(G)\) is an \emph{edge detour basis} of \(G\). A connected graph \(G\) is called an \emph{edge detour graph} if it has an edge detour set. Certain general properties of these concepts are studied. The edge detour numbers of certain classes of graphs are determined. We show that for each pair of integers \(k\) and \(p\) with \(2 \leq k < p\), there is an edge detour graph \(G\) of order \(p\) with \(dn_1(G) = k\). An edge detour set \(S\), no proper subset of which is an edge detour set, is a \emph{minimal edge detour set}. The \emph{upper edge detour number} \(dn_1^+(G)\) of a graph \(G\) is the maximum cardinality of a minimal edge detour set of \(G\). We determine the upper edge detour numbers of certain classes of graphs. We also show that for every pair \(a, b\) of integers with \(2 \leq a \leq 6\), there is an edge detour graph \(G\) with \(dn_1(G) = a\) and \(dn_1^+(G) = b\).