Drawing Planar Graphs with Reduced Height (original) (raw)

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graph drawing , planar graph , plane 3-tree , polyline drawing , layered drawing , straight-line drawing

Abstract

A polyline (resp., straight-line) drawing Gamma\GammaGamma of a planar graph GGG on a set LkL_kLk of kkk parallel lines is a planar drawing that maps each vertex of GGG to a distinct point on LkL_kLk and each edge of GGG to a polygonal chain (resp., straight line segment) between its corresponding endpoints, where the bends lie on LkL_kLk. The height of Gamma\GammaGamma is kkk, i.e., the number of lines used in the drawing. In this paper we establish new upper bounds on the height of polyline drawings of planar graphs using planar separators. Specifically, we show that every nnn-vertex planar graph with maximum degree Delta\DeltaDelta, having an edge separator of size lambda\lambdalambda, admits a polyline drawing with height 4n/9+O(lambda)4n/9+O(\lambda)4n/9+O(lambda), where the previously best known bound was 2n/32n/32n/3. Since lambdainO(sqrtnDelta)\lambda\in O(\sqrt{n\Delta})lambdainO(sqrtnDelta), this implies the existence of a drawing of height at most 4n/9+o(n)4n/9+o(n)4n/9+o(n) for any planar triangulation with Deltaino(n)\Delta \in o(n)Deltaino(n). For nnn-vertex planar 333-trees, we compute straight-line drawings, with height 4n/9+O(1)4n/9+O(1)4n/9+O(1), which improves the previously best known upper bound of n/2n/2n/2. All these results can be viewed as an initial step towards compact drawings of planar triangulations via choosing a suitable embedding of the graph.

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How to Cite

Durocher, S., & Mondal, D. (2017). Drawing Planar Graphs with Reduced Height. Journal of Graph Algorithms and Applications, 21(4), 433–453. https://doi.org/10.7155/jgaa.00424

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