On the Steiner 4-Diameter of Graphs.

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 subset of V (G), the Steiner distance d(G)(S) among the vertices of S is the minimum size among all connected subgraphs whose vertex sets contain S. Let n, k be two integers with 2 <= k <= n. Then the Steiner k-eccentricity e(k)(v) of a vertex v of G is defined by e(k)(v) = max{d(S)vertical bar S subset of V(G), vertical bar S vertical bar = k, and v epsilon S}. Furthermore, the Steiner k-diameter of G is sdiam(k)(G) = max {e(k)(v) vertical bar v epsilon V (G)}. In 2011, Chartrand, Okamoto and Zhang showed that k - 1 <= sdiam(k)(G) <= n - 1. In this paper, graphs with sdiam(4)(G) = 3, 4, n - 1 are characterized, respectively.