## Abstract

For any fixed parameter k fi 1, a tree k-spanner of a graph G is a spanning tree T in G such that the distance between every pair of vertices in T is at most k times their distance in G. In this paper, we generalize on this very restrictive concept, and introduce Steiner tree k-spanners: We are given an input graph consisting of terminals and Steiner vertices, and we are now looking for a tree k-spanner that spans all terminals. The complexity status of deciding the existence of a Steiner tree k- spanner is easy for some k: it is NP-hard for k fi 4, and it is in P for k = 1. For the case k = 2, we develop a model in terms of an equivalent tree covering problem, and use this to show NP-hardness. By showing the NP-hardness also for the case k = 3, the complexity results for all k are complete. We also consider the problem of finding a smallest Steiner tree k-spanner (if one exists at all). For any arbitrary k fi 2, we prove that we cannot hope to find eficiently a Steiner tree k-spanner that is closer to the smallest one than within a logarithmic factor. We conclude by discussing some problems related to the model for the case k = 2.

Original language | English |
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Title of host publication | Graph-Theoretic Concepts in Computer Science - 26th International Workshop, WG 2000 Konstanz, Germany, June 15-17, 2000 Proceedings |

Editors | Ulrik Brandes, Dorothea Wagner |

Publisher | Springer Verlag |

Pages | 206-217 |

Number of pages | 12 |

ISBN (Print) | 3540411836 |

DOIs | |

State | Published - 2000 |

Event | 26th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2000 - Konstanz, Germany Duration: 15 Jun 2000 → 17 Jun 2000 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 1928 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 26th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2000 |
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Country/Territory | Germany |

City | Konstanz |

Period | 15/06/00 → 17/06/00 |

### Bibliographical note

Publisher Copyright:© Springer-Verlag Berlin Heidelberg 2000.