Upward Embedding Research Articles

How to Fit a Tree in a Box

We study compact straight-line embeddings of trees. We show that perfect binary trees can be embedded optimally: a tree with n nodes can be drawn on a \(\sqrt{n}\) by \(\sqrt{n}\) grid. We also show that testing whether a given rooted binary tree has an upward embedding with a given combinatorial embedding in a given grid is NP-hard.

Radial Level Planarity with Fixed Embedding

We study level planarity testing of graphs with a fixed combinatorial embedding for three different notions of combinatorial embeddings, namely the level embedding, the upward embedding and the planar embedding. These notions allow for increasing degrees of freedom in their corresponding drawings. For the fixed level embedding there are known and easy to test level planarity criteria. We use these criteria to prove an "untangling" lemma that plays a key role in a simple level planarity test for the case where only the upward embedding is fixed. This test is then adapted to the case where only the planar embedding is fixed. Further, we characterize radial upward planar embeddings, which lets us extend our results to radial level planarity. No algorithms were previously known for these problems.

Extending upward planar graph drawings

In this paper we study the computational complexity of the Upward Planarity Extension problem, which takes as input an upward planar drawing ΓH of a subgraph H of a directed graph G and asks whether ΓH can be extended to an upward planar drawing of G. Our study fits into the line of research on the extensibility of partial representations, which has recently become a mainstream in Graph Drawing.We show the following results.–First, we prove that the Upward Planarity Extension problem is NP-complete, even if G has a prescribed upward embedding, the vertex set of H coincides with the one of G, and H contains no edge.–Second, we show that the Upward Planarity Extension problem can be solved in O(nlog⁡n) time if G is an n-vertex upward planar st-graph. This result improves upon a known O(n2)-time algorithm, which however applies to all n-vertex single-source upward planar graphs.–Finally, we show how to solve in polynomial time a surprisingly difficult version of the Upward Planarity Extension problem, in which the underlying graph of G is a path or a cycle, G has a prescribed upward embedding, H contains no edges, and no two vertices share the same y-coordinate in ΓH.

On the sphericity testing of single source digraphs

A digraph D = ( V , A ) is called spherical, if it has an upward embedding on the round sphere which is an embedding of D on the round sphere so that all edges are monotonic arcs and all point to a fixed direction, say to the north pole. It is easy to see that [S.M. Hashemi, Digraph embedding, Discrete Math. 233 (2001) 321–328] for upward embedding, plane and sphere are not equivalent, which is in contrast with the fact that they are equivalent for undirected graphs. On the other hand, it has been proved that sphericity testing for digraphs is an NP- complete problem [ S.M. Hashemi, A. Kisielewicz, I. Rival, The complexity of upward drawings on spheres, Order 14 (1998) 327–363]. In this work we study sphericity testing of the single source digraphs. In particular, we shall present a polynomial time algorithm for sphericity testing of three connected single source digraphs.

Upward Planar Drawings and Switch-regularity Heuristics

In this paper we present a new characterization of switch-regular upward embeddings, a concept introduced by Di Battista and Liotta in 1998. This characterization allows us to deflne a new e‐cient algorithm for computing upward planar drawings of embedded planar digraphs. If compared with a popular approach described by Bertolazzi, Di Battista, Liotta, and Mannino, our algorithm computes drawings that are signiflcantly better in terms of total edge length and aspect ratio, especially for low-density digraphs. Also, we experimentally prove that the running time of the drawing process is reduced in most cases.

Upward Embeddings and Orientations of Undirected Planar Graphs

An upward embedding of an embedded planar graph states, for each vertex v, which edges are incident to v above or below and, in turn, induces an upward orientation of the edges. In this paper we characterize the set of all upward embeddings and orientations of a plane graph by using a simple flow model. We take advantage of such a flow model to compute upward orientations with the minimum number of sources and sinks of 1-connected graphs. Our theoretical results allow us to easily compute visibility representations of 1-connected graphs while having a certain control over the width and the height of the computed drawings, and to deal with partial assignments of the upward embeddings underlying the visibility representations.

Digraph embedding

An upward embedding of a digraph (directed graph) on the plane or a surface is an embedding of its underlying graph so that all directed edges are monotonic and point to a fixed direction. Such embedding in some literatures is called upward drawing without crossing of edges. For a given digraph G to decide whether it has an upward embedding on the plane is known as an NP-Complete problem (cf. [6,4]). In this paper we study the problem of upward embedding of digraphs on the round sphere. We shall present a characterization of all spherical digraphs.