The analysis of large graphs plays a prominent role in various fields of research and is relevant in many important application areas. Effective visual analysis of graphs requires appropriate visual presentations in combination with respective user interaction facilities and algorithmic graph analysis methods. How to design appropriate graph analysis systems depends on many factors, including the type of graph describing the data, the analytical task at hand and the applicability of graph analysis methods. The most recent surveys of graph visualization and navigation techniques cover techniques that had been introduced until 2000 or concentrate only on graph layouts published until 2002. Recently, new techniques have been developed covering a broader range of graph types, such as timevarying graphs. Also, in accordance with ever growing amounts of graph-structured data becoming available, the inclusion of algorithmic graph analysis and interaction techniques becomes increasingly important. In this State-of-the-Art Report, we survey available techniques for the visual analysis of large graphs. Our review first considers graph visualization techniques according to the type of graphs supported. The visualization techniques form the basis for the presentation of interaction approaches suitable for visual graph exploration. As an important component of visual graph analysis, we discuss various graph algorithmic aspects useful for the different stages of the visual graph analysis process. We also present main open research challenges in this field.

/pdf/visual-analysis-of-large-graphs-state-of-the-art-and-future-1y02vt71v3.pdf

Visual analysis of large graphs : state-of-the-art and future research challenges

The crossing number is a popular tool in graph drawing and visualization, but there is not really just one crossing number; there is a large family of crossing number notions of which the crossing number is the best known. We survey the rich variety of crossing number variants that have been introduced in the literature for purposes that range from studying the theoretical underpinnings of the crossing number to crossing minimization for visualization problems.

The Graph Crossing Number and its Variants: A Survey

If graph drawing is to become a methodological tool instead of an illustrative art, many concerns need to be overcome. We discuss the problems of social network visualization, and particularly, problems of dynamic network visualization. We consider issues that arise from the aggregation of continuous-time relational data (\"streaming\" interactions) into a series of networks. We discuss our experience developing SoNIA (Social Network Image Animator, http://sonia.stanford.edu) as a prototype platform for testing and comparing layouts and techniques, and as a tool for browsing attribute-rich network data and for animating network dynamics over time. We discuss strengths and weakness of existing layout algorithms and suggest ways to adapt them to sequential layout tasks. As such, we propose a framework for visualizing social networks and their dynamics, and we present a tool that enables debate and reflection on the quality of visualizations used in empirical research. 1 This in an equally co-authored paper (names in alphabetical order). The project was generously supported by a research incentive award given to McFarland by Stanford University's Office of Technology and Licensing (grant# 2-CDZ-108). We thank the editor and anonymous JOSS reviewers for their insightful comments, and extend special thanks to Kaisa Snellman, Ben Shaw, Ozgecan Kocak, and James Moody for their contributions at various stages of this project. We also benefited from valuable feedback from participants of James G. March’s “Monday Munch” at Stanford University and participants at the satellite symposium run by Tom Snijders, “Dynamics of Networks and Behavior,” in Portoroz, Slovenia (2004). Please send all correspondence to Daniel A. McFarland, 485 Lasuen Mall, Stanford, CA 94305 (mcfarland@stanford.edu).

The Art and Science of Dynamic Network Visualization.

Graph Drawing Beyond Planarity is a rapidly growing research area that classifies and studies geometric representations of nonplanar graphs in terms of forbidden crossing configurations. The aim of this survey is to describe the main research directions in this area, the most prominent known results, and some of the most challenging open problems.

pdf/a-survey-on-graph-drawing-beyond-planarity-590emahil6.pdf

A Survey on Graph Drawing Beyond Planarity

In a total order of the vertices of a graph, two edges with no endpoint in common can be \emphcrossing, \emphnested, or \emphdisjoint. A \emphk-stack (respectively, \emphk-queue, \emphk-arch) \emphlayout of a graph consists of a total order of the vertices, and a partition of the edges into k sets of pairwise non-crossing (non-nested, non-disjoint) edges. Motivated by numerous applications, stack layouts (also called \emphbook embeddings) and queue layouts are widely studied in the literature, while this is the first paper to investigate arch layouts.\par Our main result is a characterisation of k-arch graphs as the \emphalmost (k+1)-colourable graphs; that is, the graphs G with a set S of at most k vertices, such that G S is (k+1)-colourable.\par In addition, we survey the following fundamental questions regarding each type of layout, and in the case of queue layouts, provide simple proofs of a number of existing results. How does one partition the edges given a fixed ordering of the vertices? What is the maximum number of edges in each type of layout? What is the maximum chromatic number of a graph admitting each type of layout? What is the computational complexity of recognising the graphs that admit each type of layout?\par A comprehensive bibliography of all known references on these topics is included. \par

/pdf/on-linear-layouts-of-graphs-1egziabj5c.pdf

On linear layouts of graphs

In this paper we consider the important class of directed acyclic graphs (DAGs), and present the results of a comparative study on four popular drawing algorithms specifically developed for them. The study has been performed within a general experimental setting consisting of two large test suites of DAGs and a set of quality measures. The focus of the experiments has been the practical behavior of the algorithms with a geometric foundation compared to that of the algorithms with a topological foundation. The four algorithms exhibit various trade-offs with respect to the quality measures considered, and none of them clearly outperforms the others. Our analysis has motivated the development of a new hybrid strategy for drawing DAGs that performs quite well in practice.

Drawing directed acyclic graphs: an experimental study

A rectangle visibility representation (RVR) of a graph consists of an assignment of axis-aligned rectangles to vertices such that for every edge there exists a horizontal or vertical line of sight between the rectangles assigned to its endpoints. Testing whether a graph has an RVR is known to be NP-hard. In this paper, we study the problem of finding an RVR under the assumption that an embedding in the plane of the input graph is fixed and we are looking for an RVR that reflects this embedding. We show that in this case the problem can be solved in polynomial time for general embedded graphs and in linear time for 1-plane graphs (i.e., embedded graphs having at most one crossing per edge). The linear time algorithm uses a precise list of forbidden configurations, which extends the set known for straight-line drawings of 1-plane graphs. These forbidden configurations can be tested for in linear time, and so in linear time we can test whether a 1-plane graph has an RVR and either compute such a representation or report a negative witness. Finally, we discuss some extensions of our study to the case when the embedding is not fixed but the RVR can have at most one crossing per edge.

On Visibility Representations of Non-Planar Graphs

Lower and upper bounds for long induced paths in 3-connected planar graphs

This paper addresses a long standing, widely studied, open question: Given a planar 3-graph G (i.e., a planar graph with vertex degree at most three), what is the best computational upper bound to compute a bend-minimum planar orthogonal drawing of G in the variable embedding setting? In this setting the algorithm can choose among the exponentially many planar embeddings of G the one that leads to an orthogonal drawing with the minimum number of bends. We answer the question by describing a linear-time algorithm that computes a bend-minimum planar orthogonal drawing of G. Also, if G is not K4, the drawing has at most one bend per edge. The existence of an orthogonal drawing Γ of a planar 3-graph such that Γ has the minimum number of bends and at most one bend per edge was previously unknown.

Optimal orthogonal drawings of planar 3-graphs in linear time

A planar orthogonal drawing $\Gamma$ of a planar graph $G$ is a geometric representation of $G$ such that the vertices are drawn as distinct points of the plane, the edges are drawn as chains of horizontal and vertical segments, and no two edges intersect except at their common end-points. A bend of $\Gamma$ is a point of an edge where a horizontal and a vertical segment meet. $\Gamma$ is bend-minimum if it has the minimum number of bends over all possible planar orthogonal drawings of $G$. This paper addresses a long standing, widely studied, open question: Given a planar 3-graph $G$ (i.e., a planar graph with vertex degree at most three), what is the best computational upper bound to compute a bend-minimum planar orthogonal drawing of $G$ in the variable embedding setting? In this setting the algorithm can choose among the exponentially many planar embeddings of $G$ the one that leads to an orthogonal drawing with the minimum number of bends. We answer the question by describing an $O(n)$-time algorithm that computes a bend-minimum planar orthogonal drawing of $G$ with at most one bend per edge, where $n$ is the number of vertices of $G$. The existence of an orthogonal drawing algorithm that simultaneously minimizes the total number of bends and the number of bends per edge was previously unknown.

Optimal Orthogonal Drawings of Planar 3-Graphs in Linear Time

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