Orthogonal point-set embeddings of 3-connected and 4-connected planar graphs

An orthogonal point-set embedding of a planar graph G on a set S of points in Euclidean plane is a drawing of G where each vertex of G is placed on a point of S, each edge is drawn as a sequence of alternate horizontal and vertical line segments and any two edges do not cross except at their common end. In this paper, we devise an algorithm for orthogonal point-set embedding of 3-connected cubic planar graphs having a hamiltonian cycle with at most (5n over 2 + 2) bends, where n is the number of vertices in G. We also give an algorithm for finding an orthogonal point-set embedding of 4-connected planar graphs with at most 6n bends. Both the algorithms run in linear time. To the best of our knowledge this is the first work on orthogonal point-set embeddings with fewer bends.