Drawing Trees with Perfect Angular Resolution and Polynomial Area

We study methods for drawing trees with perfect angular resolution, i.e., with angles at each node v equal to 2{\pi}/d(v). We show: 1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution. 3. Any ordered tree has a crossing-free Lombardi-style drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area. Thus, our results explore what is achievable with straight-line drawings and what more is achievable with Lombardi-style drawings, with respect to drawings of trees with perfect angular resolution.Comment: 30 pages, 17 figure

Computing upward topological book embeddings of upward planar digraphs

This paper studies the problem of computing an upward topological book embedding of an upward planar digraph G, i.e. a topological book embedding of G where all edges are monotonically increasing in the upward direction. Besides having its own inherent interest in the theory of upward book embeddability, the question has applications to well studied research topics of computational geometry and of graph drawing. The main results of the paper are as follows. -Every upward planar digraph G with n vertices admits an upward topological book embedding such that every edge of G crosses the spine of the book at most once. -Every upward planar digraph G with n vertices admits a point-set embedding on any set of n distinct points in the plane such that the drawing is upward and every edge of G has at most two bends. -Every pair of upward planar digraphs sharing the same set of n vertices admits an upward simultaneous embedding with at most two bends per edge

Simultaneous Graph Embedding with Bends and Circular Arcs

We consider the problem of simultaneous embedding of planar graphs. We demonstrate how to simultaneously embed a path and an n-level planar graph and how to use radial embeddings for curvilinear simultaneous embeddings of a path and an outerplanar graph. We also show how to use star-shaped levels to find 2-bends per path edge simultaneous embeddings of a path and an outerplanar graph. All embedding algorithms run in O(n) time

Simultaneous graph embedding with bends and circular arcs

Abstract. We consider the problem of simultaneous embedding of planar graphs. We demonstrate how to simultaneously embed a path and an n-level planar graph and how to use radial embeddings for curvilinear simultaneous embeddings of a path and an outerplanar graph. We also show how to use star-shaped levels to find 2-bends per path edge simultaneous embeddings of a path and an outerplanar graph. All embedding algorithms run in O(n) time.