Geometry of planar graphs with angles

t. Introduction: In this paper, we conslder certain geometric problems regarding planar graphs. We are interested in straight line embeddings of planar graphs that satisfy certain angle constraints. Thus this area is a blend of ~ n a r graph theory and plane geometry. The main problem discussed in this paper is the problem of planarity of angle-graphs. An angle-graph is an undirected graph in which at each vertex we are given a cyclic ordering of the incident edges as well as the angles between the successive edges. We are interested in the question of when such angle-graphs have a straight planar embedding, in which the angles between the edges at each vertex are preserved. In this paper we explain some partial results regarding the characterization of the planarity of angle-graphs, and pose some conjectures and open questions. In particular, we describe some necessary conditions for the planarity of general angle-graphs, and give necessary and sufficient conditions for the planarity of some special classes of angle-graphs.