Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems

In this paper we extend the concept of the regular edge labeling for general plane graphs and for triconnected triangulated plane graphs to 4-connected triangulated plane graphs. We present two different linear time algorithms for constructing such a labeling. By using regular edge labeling, we present a new linear time algorithm for constructing rectangular dual of planar graphs. Our algorithm is simpler than previously known algorithms. The coordinates of the rectangular dual constructed by our algorithm are integers, while the one constructed by known algorithms are real numbers. Our second regular edge labeling algorithm is based on canonical ordering of 4-connected triangulated plane graphs. By using this technique, we present a new algorithm for constructing visibility representation of 4-connected planar graphs. Our algorithm reduces the size of the representation by a factor of 2 for such graphs.