Straight Line Representations of Infinite Planar Graphs

graphs are denoted by capital roman letters and plane graphs by capital greek letters. If the graph G is isomorphic to the plane graph F, then F is a representation of G. If the edges of F are polygonal arcs (respectively straight line segments), then F is a polygonal arc representation (respectively straight line representation). A closed Jordan curve partitions the Euclidean plane into a bounded region called the interior and an unbounded region called the exterior of the curve. A finite, plane graph F partitions the plane into a finite number of regions one of which is Received 4 November, 1976. Research supported in part by the Danish Natural Science Research Council. [J. LONDON MATH. SOC. (2), 16 (1977), 411-423] 412 CARSTEN THOMASSEN unbounded. If the boundary of every region is a 3-cycle, then F is a triangulation. Two finite, plane graphs F l s F2 are equivalent if there is a 1-1 incidence-preserving map taking the set of regions of I \ onto the set of regions of F2, the set of vertices of I \ onto the set of vertices of F2 , and the set of edges of I \ onto the set of edges of F2. If, furthermore, the unbounded region of I \ corresponds to the unbounded region of F 2 , then I \ and F2 are strongly equivalent. Equivalence between plane graphs is generalised in the obvious way to equivalence between infinite plane graphs with finite edge sets and vertex sets with no accumulation points. For any point p in the plane and any positive real number e, we denote by D(p, e), C(p, e) and D(j>, e) the set of points of (Euclidean) distance less than e, equal to e, and less than or equal to e, respectively. If p and q are points, then [p, q] denotes the straight line segment between p and q. The point set of a plane graph F is denoted by F*. If F is a plane graph and p is a point not in F*, then p is admissible with respect to F if and only if for every vertex q of F adjacent to the region containing p we have: [q, p] n F* = {q}. A plane graph F is star-shaped if every region contains an admissible point. We need the following standard results about finite planar graphs: (1). If G is a graph containing a vertex v such that G — v is a cycle, then any two representations of G are equivalent. (2). The boundary of any region of a 2-connected finite, plane graph is a cycle of