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The Steiner distance of a graph, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph G of order at least 2 and S ⊆ V (G), the Steiner distance d(S) among the vertices of S is the minimum size among all connected subgraphs whose vertex sets contain S. Let n and k be two integers with 2 ≤ k ≤ n. Then the Steiner k-eccentricityIn 2011, Chartrand, Okamoto, Zhang showed that k − 1 ≤ sdiam k (G) ≤ n − 1. In this paper, graphs with sdiam k (G) = ℓ for k = n, n − 1, n − 2, n − 3 and k − 1 ≤ ℓ ≤ n − 1 are characterized, respectively.

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