Trémaux Trees and Planarity

We present two characterizations of planarity based on Trémaux trees (i.e. DFS trees). From the last one, we deduce a simple and efficient planarity test algorithm which is eventually a new implementation of the Hopcroft Tarjan planarity algorithm. We finally recall a theorem on “cotree critical non-planar graphs” which very much simplify the search of a Kuratowski subdivision in a non-planar graph.