## 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.

Original language | English |
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Pages (from-to) | 5298-5307 |

Number of pages | 10 |

Journal | Discrete Mathematics |

Volume | 308 |

Issue number | 22 |

DOIs | |

State | Published - 28 Nov 2008 |

## Keywords

- Block graph
- Steiner center
- Steiner distance
- Steiner median
- Steiner n-distance
- Steiner n-eccentricity