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For a nonempty subset \(S\) of vertices of a \(k\)-connected graph \(G\) and for \(1 \leq i \leq k\), the Steiner \(i\)-distance \(d_i(S)\) of \(S\) is the minimum size among all \(i\)-connected subgraphs containing \(S\). Relationships between Steiner \(i\)-distance and the connectivity and hamiltonian properties of a graph are discussed. For a \(k\)-connected graph \(G\) of order \(p\) and integers \(i\) and \(n\) with \(1 \leq i \leq k\) and \(1 \leq n \leq p\), the \((i, n)\)-eccentricity of a vertex \(v\) of \(G\) is the maximum Steiner \(i\)-distance \(d_i(S)\) of a set \(S\) containing \(v\) with \(|S| = n\). The \((i, n)\)-center \(C_{i,n}(G)\) of \(G\) is the subgraph induced by those vertices with minimum \((i, n)\)-eccentricity. It is proved that for every graph \(H\) and integers \(i,n \geq 2\), there exists an \(i\)-connected graph \(G\) such that \(C_{i,n}(G) \cong H\).