Steiner diameter of 3, 4 and 5-connected maximal planar graphs

Let G be a connected graph of order p and S a nonempty set of vertices of G. Then the Steiner distance d(S) of S is the minimum size of a connected subgraph of G whose vertex set contains S. If n is an integer, 2 ≤ n ≤ p, the Steiner n-diameter, diamn(G), of G is the maximum Steiner distance of any n-subset of vertices of G. This is a generalisation of the ordinary diameter, which is the case n = 2. We give upper bounds on the Steiner n-diameter of maximum planar graphs in terms of order and connectivity. Moreover, we construct graphs to show that the bound is asymptotically sharp. Furthermore we extend this result to 4 and 5-connected maximal planar graphs.