Algorithms for area-efficient orthogonal drawings

Abstract

An orthogonal drawing of a graph is a drawing such that vertices are placed on grid points and edges are drawn as sequences of vertical and horizontal segments. In this paper we present linear time algorithms that produce orthogonal drawings of graphs with n vertices. If the maximum degree is four, then the drawing produced by our first algorithm needs area at most (roughly) 0.76 n 2, and introduces at most 2 n + 2 bends. Also, each edge of such a drawing has at most two bends. Our algorithm is based on forming and placing pairs of vertices of the graph. If the maximum degree is three, then the drawing produced by our second algorithm needs at most (roughly) ( 1 4 )n 2 area and, if the graph is biconnected, at most ⌊ n 2 ⌋ + 3 bends. These upper bounds match the upper bounds known for planar graphs of maximum degree 3. This algorithm produces optimal drawings (within a constant of 2) with respect to the number of bends, since there is a lower bound of n 2 + 1 in the number of bends for orthogonal drawings of maximum degree 3 graphs.