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Spirality and Optimal Orthogonal Drawings

We deal with the problem of constructing the orthogonal drawing of a graph with the minimum number of bends along the edges. The problem has been recently shown to be NP-complete in the general case. In this paper we introduce and study the new concept of spirality, which is a measure of how an orthogonal drawing is "rolled up," and develop a theory on the interplay between spirality and number of bends of orthogonal drawings. We exploit this theory to present polynomial time algorithms for two significant classes of graphs: series-parallel graphs and 3-planar graphs. Series-parallel graphs arise in a variety of problems such as scheduling, electrical networks, data-flow analysis, database logic programs, and circuit layout. Also, they play a central role in planarity problems. Furthermore, drawings of 3-planar graphs are a classical field of investigation.