The minimum Steiner k-eccentricity

Given a connected graph G=(V,E) and a vertex set S⊂ V, the Steiner distance d(S) of S is the size of a minumum spanning tree of S in G. For a connected graph G of order n and an integer k with 2≤ k ≤ n, the Steiner k-eccentricity of v of a vertex v in G is the maximum value of d(S) over all S⊂ V with |S|=k and v∈ S. The minimum Steiner k-eccentricity, srad_k(G), is called the Steiner k-radius of G and the maximum Steriner k-eccentricity, sdiam_k(G), is called the Steiner k-diameter of G. In 1990, Henning, Oellermann, and Swart showed that for any integer k≤ 2, there exists a graph G__k such that sdiam__k(G__k)=(2(k+1))/(2k-1)srad_k(G_k) and proved that sdiam₃(G)≤ 8/5srad₃(G), and sdiam₄(G)≤ 10/7srad₃(G) for all connected graphs G. In this talk, we will show that for each k≤ 5, sdiam_k(G)≤ (k+3)/(k+1)srad_k(G) and show that this bound is tight via a construction.