Monotone Drawings of Graphs with Fixed Embedding
A drawing of a graph is a monotone drawing if for every pair of vertices u and v there is a path drawn from u to v that is monotone in some direction. In this paper we investigate planar monotone drawings in the fixed embedding setting, i.e., a planar embedding of the graph is given as part of the input that must be preserved by the drawing algorithm. In this setting we prove that every planar graph on n vertices admits a planar monotone drawing with at most two bends per edge and with at most 4n−10 bends in total; such a drawing can be computed in linear time and requires polynomial area. We also show that two bends per edge are sometimes necessary on a linear number of edges of the graph. Furthermore, we investigate subclasses of planar graphs that can be realized as embedding-preserving monotone drawings with straight-line edges. In fact, we prove that biconnected embedded planar graphs and outerplane graphs always admit such drawings, and describe linear-time drawing algorithms for these two graph classes.
KeywordsMonotone drawings Fixed embedding Planar graph drawing Polynomial area Curve complexity
We thank the anonymous reviewers for their valuable comments.
- 3.Arkin, E.M., Connelly, R., Mitchell, J.S.B.: On monotone paths among obstacles with applications to planning assemblies. In: Symposium on Computational Geometry, pp. 334–343 (1989) Google Scholar
- 6.Brocot, A.: Calcul des rouages par approximation, nouvelle methode. Rev. Chronom. 6, 186–194 (1860) Google Scholar
- 13.Huang, W., Eades, P., Hong, S.-H.: A graph reading behavior: geodesic-path tendency. In: PacificVis, pp. 137–144 (2009) Google Scholar