Steiner centers and Steiner medians of graphs

Abstract

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.