Name
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 d Π denote the distance functions in Γ and in Π, respectively. One calls Π a quasi-axis if lim inf d ( x , y )/ d Π ( x , y ) > 0, where x and y are vertices of Π and d Π ( 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. |
Year | DOI | Venue |
|---|---|---|
1995 | 10.1016/0012-365X(94)00054-M | Discrete Mathematics |
Keywords | Field | DocType |
double ray,finite planar graph,distance function,planar graph | Discrete mathematics,Graph,Combinatorics,Vertex (geometry),Planar embedding,Planar,Mathematics,Planar graph,Bounded function | Journal |
Volume | Issue | ISSN |
145 | 1-3 | Discrete Mathematics |
Citations | PageRank | References |
6 | 0.80 | 4 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| C. Paul Bonnington | 1 | 100 | 19.95 |
| Wilfried Imrich | 2 | 444 | 53.81 |
| Mark E. Watkins | 3 | 109 | 32.53 |