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For a connected graph \(G\) of order \(p \geq 2\), a set \(S \subseteq V(G)\) is a geodetic set of \(G\) if each vertex \(v \in V(G)\) lies on an \(x\)-\(y\) geodesic for some elements \(x\) and \(y\) in \(S\). The minimum cardinality of a geodetic set of \(G\) is defined as the geodetic number of \(G\), denoted by \(g(G)\). A geodetic set of cardinality \(g(G)\) is called a \(g\)-set of \(G\). A connected geodetic set of \(G\) is a geodetic set \(S\) such that the subgraph \(G[S]\) induced by \(S\) is connected. The minimum cardinality of a connected geodetic set of \(G\) is the connected geodetic number of \(G\) and is denoted by \(g_c(G)\). A connected geodetic set of cardinality \(g_c(G)\) is called a \(g_c\)-set of \(G\). Connected graphs of order \(p\) with connected geodetic number \(2\) or \(p\) are characterized. It is shown that for positive integers \(r,d\) and \(n \geq d+1\) with \(r \leq d \leq 2r\), there exists a connected graph \(G\) of radius \(r\), diameter \(d\) and \(g_c(G) = n\). Also, for integers \(p,d\) and \(n\) with \(2 \leq d \leq p-1\), \(d+1 \leq n \leq p\), there exists a connected graph \(G\) of order \(p\), diameter \(d\) and \(g_c(G) = n\).