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

Let GG be a connected graph of order pp and SS a nonempty set of vertices of GG. Then the Steiner distance d(S)d(S) of SS is the minimum size of a connected subgraph of GG whose vertex set contains SS. If nn is an integer, 2≤n≤p2≤n≤p, the Steiner nn-diameter, diamn(G), of GG is the maximum Steiner distance of any nn-subset of vertices of GG. This is a generalisation of the ordinary diameter, which is the case n=2n=2. We give upper bounds on the Steiner nn-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.