On topological aspects of orientations

1. Introduction Constrained orientations, that is orientations such that all the vertices have a prescribed indegree, relates to one another many combinatorial and topological properties such as arboricity, connectivity and planarity. These orientations are the basic tool to solve planar augmentation problems 2]. We are concerned with two classes of planar graphs: maximal planar graphs (i.e. polyhedral graphs, triangulations) and maximal bipartite planar graphs (i.e. bipartite planar graphs with quadrilateral faces). For these graphs we consider constrained orientations with a constant indegree for the internal vertex set. We recall or prove new fundamental relations between these orientations, speciic tree decompositions and bipolar orientations. In particular, these relations yield linear time computation algorithms. Using these orientations, we give a characterization of 4-connected maximal planar graphs and 3-connected planar graphs, which lead to very simple linear time algorithms.