We prove that every n-vertex graph G with pathwidth pw(G) has a three-dimensional straight-line grid drawing with O(pw(G) n) volume. Thus for

In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A k-stack (resp...

Spring algorithms are regarded as effective tools for visualizing undirected graphs. One major feature of applying spring algorithms is to display symmetric properties of graphs. This feature has been confirmed by numerous experiments. In this paper, firstly we formalize the concepts of graph symmetries in terms of "reflectional" and "rotational" automorphisms; and characterize the types of symmetries, which can be displayed simultaneously by a graph layout, in terms of "geometric" automorphism groups. We show that our formalization is complete. Secondly, we provide general theoretical evidence of why many spring algorithms can display graph symmetry. Finally, the strength of our general theorem is demonstrated from its application to several existing spring algorithms. 1 Introduction Graphs are commonly used in Computer Science to model relational structures such as programs, databases, and data structures. A good graph "layout" gives a clear understanding of a structural model; a ba...

In this paper we investigate the general position model for the drawing of arbitrary degree graphs in the D-dimensional (D >= 2) orthogonal grid. In this model no two vertices lie in the same grid hyperplane.

Let G be an n-node graph. We address the problem of computing a maximum symmetric graph H from G by deleting nodes, deleting edges, and contracting edges. This NP-complete problem arises naturally from the objective of drawing G as symmetrically as possible. We show that its tractability for the special cases of G being a plane graph, an ordered tree, and an unordered tree, depends on the type of operations used to obtain H from G. Moreover, we give an O(log n)-approximation algorithm for the intractable case that H is obtained from a tree G by contracting edges. As a by-product, we give an O(log n)-approximation algorithm for an NP-complete edit-distance problem.

Abstract. Symmetry is one of the most important aesthetic criteria in graph drawing because it reveals structure in the graph. This paper discusses symmetric drawings of oneconnected planar graphs. More specifically, we discuss planar (geometric) automorphisms, that is, automorphisms of a graph G that can be represented as symmetries of a planar drawing of G. Finding planar automorphisms is the first and most difficult step in constructing planar symmetric drawings of graphs. The problem of determining whether a given graph has a nontrivial geometric automorphism is NP-complete for general graphs. The two previous papers in this series have discussed the problem of drawing planar graphs with a maximum number of symmetries, for the restricted cases where the graph is triconnected and biconnected. This paper extends the previous results to cover planar graphs that are oneconnected. We present a linear time algorithm for drawing oneconnected planar graphs with a maximum number of symmetries. Key Words.

For trees, we dene the notion of the so-called symmetry number to measure the size of the maximum subtree that exhibits an axial symmetry in graph drawing. For unrooted unordered trees, we are able to demonstrate a polynomial time algorithm for computing the symmetry number. Keywords: Design of Algorithms, Graph Drawing. 1. Introduction Graphs are known to be useful for modeling various scientic/engineering problems in the real world. Because of the popularity of graphs, graph drawing has emerged as a research topic of great importance in graph theory. In many cases, a `pretty drawing' often oers more insights into the nature of a graph. A natural question arises: How to dene `pretty drawings?' Aesthetic guidelines suggested in the literature (see, e.g., [1, 8]) for drawing pretty graphs include minimizing the number of edge crossings, minimizing the variance of edge lengths, minimizing the number of bends, as well as drawing edges orthogonally or using straight-line segments. Su...

Abstract: The proper visualization and monitoring of their (ongoing) business processes is crucial for any enterprise. Thus a broad spectrum of processes has to be visualized ranging from simple, short–running processes to complex long–running ones (consisting of up to hundreds of activities). In any case, users shall be able to quickly understand the logic behind a process and to get a quick overview of related tasks. One practical problem arises when different fragments of a business process are scattered over several systems where they are often modeled using different process meta models (e.g., High–Level Petri Nets). The challenge is to find an integrated and user–friendly visualization for these business processes. In this paper we discover use cases relevant in this context. Since existing graph layout approaches have focused on general graph drawing so far we further develop a specific approach for layouting business process graphs. The work presented in this paper is embedded within a larger project (Proviado) on the visualization of automotive processes. 1

We propose a force-directed approach for drawing graphs in a nearly symmetric fashion. Our algorithm is built upon recent theoretical results on maximum symmetric subgraphs. Knowing the sequence of edge contractions sufficient for turning an asymmetric graph into a symmetric subgraph, our approach in symmetric drawing begins by drawing a graph's maximum symmetric subgraph using a force-directed method first; then the contracted edges are re-inserted back into the drawing. Considering symmetry as the underlying aesthetic criterion, our algorithm provides better drawings than the conventional spring algorithms, as our experimental results indicate.

A track layout of a graph consists of a vertex colouring, an edge colouring, and a total order of each vertex colour class such that between each pair of vertex colour classes, there is no monochromatic pair of crossing edges. This paper studies track layouts and their applications to other models of graph layout. In particular, we improve on the results of Enomoto and Miyauchi [SIAM J. Discrete Math., 1999] regarding stack layouts of subdivisions, and give analogous results for queue layouts. We solve open problems due to Felsner, Wismath, and Liotta [Proc. Graph Drawing, 2001] and Pach, Thiele, and Toth [Proc. Graph Drawing, 1997] concerning three-dimensional straight-line grid drawings. We initiate the study of three-dimensional polyline grid drawings and establish volume bounds within a logarithmic factor of optimal.