Nodally 3-connected planar graphs and convex combination mappings

A barycentric mapping of a planar graph is a plane embedding in which every internal vertex is the average of its neighbours. A celebrated result of Tutte’s [16] is that if a planar graph is nodally 3-connected then such a mapping is an embedding. Floater generalised this result to convex combination mappings in which every internal vertex is a proper weighted average of its neighbours. He also generalised the result to all triangulated planar graphs. This has applications in numerical analysis (grid generation), and in computer graphics (image morphing, surface triangulations, texture mapping): see [6, 17]. White [17] showed that every chord-free triangulated planar graph is nodally 3connected. We show that (i) a nontrivial plane embedded graph is nodally 3-connected if and only if every face boundary is a simple cycle and the intersection of every two faces is connected; (ii) every convex combination mapping of a plane embedded graph G is an embedding if and only if (a) every face boundary is a simple cycle, (b) the intersection of every two bounded faces is connected, and (c) there are no so-called inverted subgraphs; (iii) this is equivalent to G admitting a convex embedding (see [13]); and (iv) any two such embeddings (with the same orientation) are isotopic. 1 Planar graphs and nodal 3-connectivity We follow the usual definitions of graphs, including paths, simple paths, cycles, simple cycles, and connectivity: [9] is a useful source on the subject. The accepted definition of graph does not allow self-loops nor multiple edges nor infinite sets of vertices, so it is a finite simple graph in Tutte’s language [16], 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 . Two vertices u, v are adjacent or neighbours if {u, v} ∈ E. Given G = (V,E), when u is considered to be a vertex, u ∈ G means u ∈ V , and when e is considered to be an edge, e ∈ G means e ∈ E. ∗These results were presented at the fourth Irish MFCSIT conference, Cork, Ireland, August 2006.