Let G be connected graph and S a set of vertices of G. Then a Steiner tree for S is a connected subgraph of G of smallest size (number of edges) that contains S. The size of such a subgraph is called the Steiner distance for S and is denoted by d(S). For a vertex of G, and integer n, 2 ≤ n ≤ |V (G)|, the Steiner n-eccentricity en(ν) of ν is defined as en(ν) = max{d(S)|S ⊆V(G),|S| = n, and ν ∈ S}. The Steiner n-radius rad nG and Steiner n-diameter diamnG are defined as the minimum and maximum n-eccentricity respectively, taken over all vertices of G. Relationships between radnG and diamnG are given if G is a tree, and a conjecture (with some supporting results) is made that relates these parameters for general graphs. The subgraph induced by these vertices with n-eccentricity radnG is called the Steiner n-center of G and is denoted by Cn(G). It is shown that every graph is the Steiner n-center of some graph. The Steiner n-distance of a vertex ν, denoted by dn(ν), is defined by dn(ν) = Σ{d(S)|ν ∈ S, |S| = n}. The Steiner n-median Mn(G) of G is the subgraph induced by those vertices with minimum Steiner n-distance. Algorithms for finding Cn(T) and Mn(T) for a tree T are described. It is shown that the distance between the Cn(T) and Mn(T) for a tree T can be arbitrarily large. Eccentricity measures are defined that extend those of the Steiner n-eccentricity and Steiner n-distance of a vertex. Then it is shown that every vertex on a shortest path between the Steiner n-center and Steiner n-median of a tree belongs to a ‘center’ relative to one of these eccentricity measures.
