Nodally 3-connected planar graphs and barycentric embeddings

An interesting question about planar graphs is whether they admit plane embeddings in which every bounded face is convex. Stein [10] gave as a necessary and sufficient condition that every face boundary be a simple cycle and every two bounded faces meet in a connected set, with an extra condition about the number of vertices on the outer face. Tutte [12] gave a similar characterisation, and later [13] showed that every nodally 3-connected planar graph admits a barycentric embedding. Floater [4] generalised this to convex combination mappings of triangulated graphs. White [14] showed that a chord-free triangulated graph is nodally 3-connected and showed that Tutte’s result applies to all triangulated graphs. We extend Tutte’s results beyond the class of triangulated graphs. We show that a biconnected plane-embedded graph is nodally 3-connected if and only if the intersection of any two faces, bounded or otherwise, is connected. If a plane embedded graph admits a convex embedding, then every face boundary is a simple cycle, the intersection of every two faces is connected, and there are no inverted subgraphs (as defined in the paper). Such graphs we call admissible. The idea of admissible embedded graph is more useful than Stein’s criterion [10] and simpler than Tutte’s [12]. We show that every admissible plane embedded graph admits a barycentric embedding. It follows immediately that a plane embedded graph has a convex embedding if and only if every barycentric map is an embedding. Finally we show that when a plane embedded graph admits a barycentric embedding, the two embeddings are isotopic. 1 Criterion for nodally 3-connected planar graphs We follow the usual definitions of graphs, paths, cycles, connectivity, plane embeddings, and planar graphs: [6] is a useful source on the subject. The accepted definition of graph does not allow selfloops nor multiple edges nor infinite sets of vertices, so it is a finite simple graph in Tutte’s language [13], and a graph G can be specified as a pair (V, E) giving its vertices and edges. E is a set of unordered pairs of distinct vertices in V . e-mail: [email protected]. Mathematics department website: http://www.maths.tcd.ie.