Separating double rays in locally finite planar graphs

✍ Scribed by C Paul Bonnington; Wilfried Imrich; Mark E Watkins

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 765 KB
• Volume: 145
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(94)00054-m

✦ Synopsis

The main aim of this paper is to characterize infinite, locally finite, planar, l-ended graphs by means of path separation properties. Let r be an infinite graph, let I7 be a double ray in r, and let d and d, denote the distance functions in r and in n, respectively. One calls II a quasi-axis if lim inf d(x, y)/d,(x, y) > 0, where x and y are vertices of II and d,(x, y) + CC. An infinite, locally finite, almost 4-connected, almost-transitive, l-ended graph is shown to be planar if and only if the complement of every quasi-axis has exactly two infinite components.

Let r be locally finite, planar, 3-connected, almost-transitive, and l-ended. It is shown that no proper planar embedding of r has an infinite face and hence its covalences are bounded. If r has bounded covalences and if ll is any double ray in r, it is shown that r -ll has at most two infinite components, at most one on each side of Il. If, moreover, 17 is a quasi-axis, then r -n 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, l-ended graphs is then obtained.