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Geodetic rays and fibers in periodic graphs
Author(s) -
Niemeyer Peter,
Watkins Mark E.
Publication year - 2000
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/(sici)1097-0118(200005)34:1<67::aid-jgt7>3.0.co;2-v
Subject(s) - geodetic datum , mathematics , combinatorics , abelian group , bounded function , graph , discrete mathematics , geodesy , mathematical analysis , geology
Using the notion of fibers, where two rays belong to the same fiber if and only if they lie within bounded Hausdorff‐distance of one another, we study how many fibers of a graph contain a geodetic ray and how many essentially distinct geodetic rays such “geodetic fibers” must contain. A complete answer is provided in the case of locally finite graphs that admit an almost transitive action by some infinite finitely generated, abelian group. Such graphs turn out to have either finitely many or uncountably many geodetic fibers. Furthermore, with finitely many possible exceptions, each of these fibers contains uncountably many geodetic rays. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 67–88, 2000