Steiner centers and Steiner medians of graphs

AbstractLet 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 dG(S) or d(S). The Steiner n-eccentricity en(v) and Steiner n-distance dn(v) of a vertex v in G are defined as en(v)=max{d(S)| S⊆V(G), |S|=n and v∈S} and dn(v)=∑{d(S)| S⊆V(G), |S|=n and v∈S}, respectively. The Steiner n-center Cn(G) of G is the subgraph induced by the vertices of minimum n-eccentricity. The Steiner n-median Mn(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 Cn(T) is contained in Cn+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 Mn(T) is contained in Mn+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 Cn(G)⊈Cn+1(G). We also show that for each n⩾2 there is a distance–hereditary graph G such that Mn(G)⊈Mn+1(G). Despite these negative examples, we prove that if G is a block graph then Mn(G) is contained in Mn+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