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# Convex Drawings of Hierarchical Graphs in Linear Time, with Applications to Planar Graph Morphing

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## Cite As

Boris Klemz. Convex Drawings of Hierarchical Graphs in Linear Time, with Applications to Planar Graph Morphing. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ESA.2021.57

## Abstract

A hierarchical plane st-graph H can be thought of as a combinatorial description of a planar drawing Γ of a 2-connected graph G in which each edge is a y-monotone curve and each face encloses a y-monotone region (that is, a region whose intersection with any horizontal line is a line segment, a point, or empty). A drawing Γ' of H is a drawing of G such that each horizontal line intersects the same left-to-right order of edges and vertices in Γ and Γ', that is, the underlying hierarchical plane st-graph of both drawings is H. A straight-line planar drawing of a graph is convex if the boundary of each face is realized as a convex polygon. We study the computation of convex drawings of hierarchical plane st-graphs such that the outer face is realized as a prescribed polygon. Chrobak, Goodrich, and Tamassia [SoCG'96] and, independently, Kleist et al. [CGTA'19] described an idea to solve this problem in O(n^{1.1865}) time, where n is the number of vertices of the graph. Also independently, Hong and Nagamochi [J. Discrete Algorithms'10] described a completely different approach, which can be executed in O(n²) time. In this paper, we present an optimal O(n) time algorithm to solve the above problem, thereby improving the previous results by Chrobak, Goodrich, and Tamassia, Kleist et al., and by Hong and Nagamochi. Our result has applications in graph morphing. A planar morph is a continuous deformation of a graph drawing that preserves straight-line planarity. We show that our algorithm can be used as a drop-in replacement to speed up a procedure by Alamdari et al. [SICOMP'17] to morph between any two given straight-line planar drawings of the same plane graph. The running time improves from O(n^{2.1865}) to O(n²log n). To obtain our results, we devise a new strategy for computing so-called archfree paths in hierarchical plane st-graphs, which might be of independent interest.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Paths and connectivity problems
• Mathematics of computing → Graphs and surfaces
• Theory of computation → Computational geometry
• Human-centered computing → Graph drawings
##### Keywords
• convex drawing
• hierarchical graph
• graph drawing
• computational geometry
• planarity
• planar graph
• morphing
• convexity

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