Geodetic Rays and Fibers in One-Ended Planar Graphs

Abstract

A fiber in an infinite graph is an equivalence class of rays whereby two rays belong to the same fiber whenever each is contained in an n -neighborhood of the other for some n <∞. As this relation is a refinement of end-equivalence, it is of interest when applied to one-ended graphs, in particular to the class G a , a * of one-ended, 3-connected, planar graphs whose valences and covalences are finite and at least a and at least a *, respectively. Any path in a graph in G 4, 4 that uses at most ⌊ 1 2 ( ρ *( f )−2)⌋ edges of any incident face f (whose covalence is ρ *( f )) is shown to be the unique geodetic path joining its end-vertices. From this is deduced that every edge lies on a geodetic double ray, proving a conjecture of Bonnington, Imrich, and Seifter except in the presence of 3-valent vertices or 3-covalent faces. If all valences are at least 4 and all covalences are at least 6, then all Petrie walks are geodetic double rays. Basic questions concerning geodetic fibers (i.e., that contain a geodetic ray) in the graphs in G a , a * are resolved, namely: (1) how many are there and (2) are they of finite, countable, or uncountable type , i.e., is every set S of geodetic rays in the fiber that is maximal subject to no two rays in S containing a common subray finite, countable, or uncountable (respectively)? A representative result is that graphs in G 4, 6 ∪ G 5, 4 contain uncountably many geodetic fibers of finite type; furthermore, every geodetic fiber in these graphs contains at most three pairwise-disjoint geodetic rays, revealing an underlying tree-like structure when growth is exponential. In this vein, it is shown that graphs in G 4, 5 ∪ G 5, 4 admit no nonidentity bounded automorphism.