## Abstract

Let P be a double ray in an infinite graph X, and let d and d_{p} denote the distance functions in X and in P respectively. One calls P a geodesic if d(x, y) = d_{p}(x, y), for all vertices x and y in P. We give situations when every edge of a graph belongs to a geodesic or a half-geodesic. Furthermore, we show the existence of geodesics in infinite locally-finite transitive graphs with polynomial growth which are left invariant (set-wise) under "translating" automorphisms. As the main result, we show that an infinite, locally-finite, transitive, 1-ended graph with polynomial growth is planar if and only if the complement of every geodesic has exactly two infinite components.

Original language | English |
---|---|

Pages (from-to) | 12-33 |

Number of pages | 22 |

Journal | Journal of Combinatorial Theory, Series B |

Volume | 67 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 1996 |

Externally published | Yes |