Geodetic rays and fibers in one-ended planar graphs (Q1354722)

Two rays in an infinite graph are said to belong to the same fiber if each is contained in a bounded neighbourhood of the other. For this refinement of end-equivalence the following basic questions are resolved within the class \({\mathcal G}_{a,a^*}\) of one-ended, 3-connected planar graphs whose valencies and covalencies are finite and bounded below by \(a\) and \(a^*\), respectively: 1. How many geodetic fibers (i.e., fibers containing a geodetic ray) are there? 2. Are they of finite, countable, or uncountable type, i.e., is every set \(\mathcal S\) of geodetic rays in the fiber that is maximal subject to no two rays in \(\mathcal S\) containing a common subray finite, countable, or uncountable? A representative result is that graphs in \({\mathcal G}_{4,6}\cup {\mathcal G}_{5,4}\) contain uncountably many geodetic fibers of infinite type; furthermore, every geodetic fiber in these graphs contains at most 3 mutually disjoint geodetic rays, revealing an underlying tree-like structure when growth is exponential. Also, it is proved that if \(a,a^*\geq 4\) then every edge lies on a geodetic double-ray; this settles a conjecture of \textit{C. P. Bonnington, W. Imrich} and \textit{N. Seifter} [J. Comb. Theory, Ser. B 67, No. 1, 12-33, Art. No. 0031 (1996; Zbl 0856.05032)].