Steiner centers and Steiner medians of graphs

Let G be a connected graph and S a set of vertices of G. The Steiner distance of S is the smallest number of edges in a connected subgraph of G that contains S and is denoted by d_G (S) or d (S). The Steiner n-eccentricity e_n (v) and Steiner n-distance d_n (v) of a vertex v in G are defined as e_n (v) = max { d (S) |S ⊆ V (G), | S | = n and v ∈ S } and d_n (v) = ∑ { d (S) |S ⊆ V (G), | S | = n and v ∈ S }, respectively. The Steiner n-center C_n (G) of G is the subgraph induced by the vertices of minimum n-eccentricity. The Steiner n-median M_n (G) of G is the subgraph induced by those vertices with minimum Steiner n-distance. Let T be a tree. Oellermann and Tian [O.R. Oellermann, S. Tian, Steiner centers in graphs, J. Graph Theory 14 (1990) 585-597] showed that C_n (T) is contained in C_{n + 1} (T) for all n ≥ 2. Beineke et al. [L.W. Beineke, O.R. Oellermann, R.E. Pippert, On the Steiner median of a tree, Discrete Appl. Math. 68 (1996) 249-258] showed that M_n (T) is contained in M_{n + 1} (T) for all n ≥ 2. Then, Oellermann [O.R. Oellermann, On Steiner centers and Steiner medians of graphs, Networks 34 (1999) 258-263] asked whether these containment relationships hold for general graphs. In this note we show that for every n ≥ 2 there is an infinite family of block graphs G for which C_n (G) ⊈ C_{n + 1} (G). We also show that for each n ≥ 2 there is a distance-hereditary graph G such that M_n (G) ⊈ M_{n + 1} (G). Despite these negative examples, we prove that if G is a block graph then M_n (G) is contained in M_{n + 1} (G) for all n ≥ 2. Further, a linear time algorithm for finding the Steiner n-median of a block graph is presented and an efficient algorithm for finding the Steiner n-distances of all vertices in a block graph is described.