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...

• Type: Journal
• Year: 2007
• Where: CORR
• Authors: Colm Ó'Dúnlaing
• Keywords: CORR 2007, Education, Internal Vertex, Planar Graphs, Simple Cycles