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- W1970209184 abstract "The main aim of this paper is to characterize infinite, locally finite, planar, 1-ended graphs by means of path separation properties. Let Γ be an infinite graph, let Π be a double ray in Γ, and let d and dΠ denote the distance functions in Γ and in Π, respectively. One calls Π a quasi-axis if lim inf d(x, y)/dΠ(x, y) > 0, where x and y are vertices of Π and dΠ(x, y) → ∞. An infinite, locally finite, almost 4-connected, almost-transitive, 1-ended graph is shown to be planar if and only if the complement of every quasi-axis has exactly two infinite components. Let Γ be locally finite, planar, 3-connected, almost-transitive, and 1-ended. It is shown that no proper planar embedding of Γ has an infinite face and hence its covalences are bounded. If Γ has bounded covalences and if Π is any double ray in Γ, it is shown that Γ − Π has at most two infinite components, at most one on each side of Π. If, moreover, Π is a quasi-axis, then Γ − Π is shown to have exactly two infinite components. With the aid of a result of Thomassen (1992), the above-stated characterization of infinite, locally finite, planar, 1-ended graphs is then obtained." @default.
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- W1970209184 date "1995-10-01" @default.
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- W1970209184 title "Separating double rays in locally finite planar graphs" @default.
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- W1970209184 doi "https://doi.org/10.1016/0012-365x(94)00054-m" @default.
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