Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graphs arising from applications such as social network analysis, cartography, linguistics, and bioinformatics. [1]
A drawing of a graph or network diagram is a pictorial representation of the vertices and edges of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph. [2] In the abstract, all that matters is which pairs of vertices are connected by edges. In the concrete, however, the arrangement of these vertices and edges within a drawing affects its understandability, usability, fabrication cost, and aesthetics. [3] The problem gets worse if the graph changes over time by adding and deleting edges (dynamic graph drawing) and the goal is to preserve the user's mental map. [4]
Graphs are frequently drawn as node–link diagrams in which the vertices are represented as disks, boxes, or textual labels and the edges are represented as line segments, polylines, or curves in the Euclidean plane. [3] Node–link diagrams can be traced back to the 14th-16th century works of Pseudo-Lull which were published under the name of Ramon Llull, a 13th century polymath. Pseudo-Lull drew diagrams of this type for complete graphs in order to analyze all pairwise combinations among sets of metaphysical concepts. [5]
In the case of directed graphs, arrowheads form a commonly used graphical convention to show their orientation; [2] however, user studies have shown that other conventions such as tapering provide this information more effectively. [6] Upward planar drawing uses the convention that every edge is oriented from a lower vertex to a higher vertex, making arrowheads unnecessary. [7]
Alternative conventions to node–link diagrams include adjacency representations such as circle packings, in which vertices are represented by disjoint regions in the plane and edges are represented by adjacencies between regions; intersection representations in which vertices are represented by non-disjoint geometric objects and edges are represented by their intersections; visibility representations in which vertices are represented by regions in the plane and edges are represented by regions that have an unobstructed line of sight to each other; confluent drawings, in which edges are represented as smooth curves within mathematical train tracks; fabrics, in which nodes are represented as horizontal lines and edges as vertical lines; [8] and visualizations of the adjacency matrix of the graph.
Many different quality measures have been defined for graph drawings, in an attempt to find objective means of evaluating their aesthetics and usability. [9] In addition to guiding the choice between different layout methods for the same graph, some layout methods attempt to directly optimize these measures.
There are many different graph layout strategies:
Graphs and graph drawings arising in other areas of application include
In addition, the placement and routing steps of electronic design automation (EDA) are similar in many ways to graph drawing, as is the problem of greedy embedding in distributed computing, and the graph drawing literature includes several results borrowed from the EDA literature. However, these problems also differ in several important ways: for instance, in EDA, area minimization and signal length are more important than aesthetics, and the routing problem in EDA may have more than two terminals per net while the analogous problem in graph drawing generally only involves pairs of vertices for each edge.
Software, systems, and providers of systems for drawing graphs include:
graphdrawing
package (requires LuaTeX). [33] In order theory, a Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set one represents each element of as a vertex in the plane and draws a line segment or curve that goes upward from one vertex to another vertex whenever covers . These curves may cross each other but must not touch any vertices other than their endpoints. Such a diagram, with labeled vertices, uniquely determines its partial order.
In computer science, a graph is an abstract data type that is meant to implement the undirected graph and directed graph concepts from the field of graph theory within mathematics.
Force-directed graph drawing algorithms are a class of algorithms for drawing graphs in an aesthetically-pleasing way. Their purpose is to position the nodes of a graph in two-dimensional or three-dimensional space so that all the edges are of more or less equal length and there are as few crossing edges as possible, by assigning forces among the set of edges and the set of nodes, based on their relative positions, and then using these forces either to simulate the motion of the edges and nodes or to minimize their energy.
In graph theory, a branch of mathematics, the triconnected components of a biconnected graph are a system of smaller graphs that describe all of the 2-vertex cuts in the graph. An SPQR tree is a tree data structure used in computer science, and more specifically graph algorithms, to represent the triconnected components of a graph. The SPQR tree of a graph may be constructed in linear time and has several applications in dynamic graph algorithms and graph drawing.
A program structure tree (PST) is a hierarchical diagram that displays the nesting relationship of single-entry single-exit (SESE) fragments/regions, showing the organization of a computer program. Nodes in this tree represent SESE regions of the program, while edges represent nesting regions. The PST is defined for all control flow graphs.
The International Symposium on Graph Drawing (GD) is an annual academic conference in which researchers present peer reviewed papers on graph drawing, information visualization of network information, geometric graph theory, and related topics.
In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity.
In graph theory, the planarity testing problem is the algorithmic problem of testing whether a given graph is a planar graph (that is, whether it can be drawn in the plane without edge intersections). This is a well-studied problem in computer science for which many practical algorithms have emerged, many taking advantage of novel data structures. Most of these methods operate in O(n) time (linear time), where n is the number of edges (or vertices) in the graph, which is asymptotically optimal. Rather than just being a single Boolean value, the output of a planarity testing algorithm may be a planar graph embedding, if the graph is planar, or an obstacle to planarity such as a Kuratowski subgraph if it is not.
Layered graph drawing or hierarchical graph drawing is a type of graph drawing in which the vertices of a directed graph are drawn in horizontal rows or layers with the edges generally directed downwards. It is also known as Sugiyama-style graph drawing after Kozo Sugiyama, who first developed this drawing style.
In graph drawing, the angular resolution of a drawing of a graph is the sharpest angle formed by any two edges that meet at a common vertex of the drawing.
Peter D. Eades is an Australian computer scientist, an emeritus professor in the School of Information Technologies at the University of Sydney, known for his expertise in graph drawing.
In graph drawing, a circular layout is a style of drawing that places the vertices of a graph on a circle, often evenly spaced so that they form the vertices of a regular polygon.
An arc diagram is a style of graph drawing, in which the vertices of a graph are placed along a line in the Euclidean plane, with edges being drawn as semicircles in one or both of the two halfplanes bounded by the line, or as smooth curves formed by sequences of semicircles. In some cases, line segments of the line itself are also allowed as edges, as long as they connect only vertices that are consecutive along the line. Variations of this drawing style in which the semicircles are replaced by convex curves of some other type are also commonly called arc diagrams.
NodeXL is a network analysis and visualization software package for Microsoft Excel 2007/2010/2013/2016. The package is similar to other network visualization tools such as Pajek, UCINet, and Gephi. It is widely applied in ring, mapping of vertex and edge, and customizable visual attributes and tags. NodeXL enables researchers to undertake social network analysis work metrics such as centrality, degree, and clustering, as well as monitor relational data and describe the overall relational network structure. When applied to Twitter data analysis, it showed the total network of all users participating in public discussion and its internal structure through data mining. It allows social Network analysis (SNA) to emphasize the relationships rather than the isolated individuals or organizations, allowing interested parties to investigate the two-way dialogue between organizations and the public. SNA also provides a flexible measurement system and parameter selection to confirm the influential nodes in the network, such as in-degree and out-degree centrality. The software contains network visualization, social network analysis features, access to social media network data importers, advanced network metrics, and automation.
In graph drawing, an upward planar drawing of a directed acyclic graph is an embedding of the graph into the Euclidean plane, in which the edges are represented as non-crossing monotonic upwards curves. That is, the curve representing each edge should have the property that every horizontal line intersects it in at most one point, and no two edges may intersect except at a shared endpoint. In this sense, it is the ideal case for layered graph drawing, a style of graph drawing in which edges are monotonic curves that may cross, but in which crossings are to be minimized.
Dominance drawing is a style of graph drawing of directed acyclic graphs that makes the reachability relations between vertices visually apparent. In dominance drawing, vertices are placed at distinct points of the Euclidean plane and a vertex v is reachable from another vertex u if and only if both Cartesian coordinates of v are greater than or equal to the coordinates of u. The edges of a dominance drawing may be drawn either as straight line segments, or, in some cases, as polygonal chains.
In the mathematical field of graph theory, planarization is a method of extending graph drawing methods from planar graphs to graphs that are not planar, by embedding the non-planar graphs within a larger planar graph.
In graph drawing, the area used by a drawing is a commonly used way of measuring its quality.
In graph drawing styles that represent the edges of a graph by polylines, it is desirable to minimize the number of bends per edge or the total number of bends in a drawing. Bend minimization is the algorithmic problem of finding a drawing that minimizes these quantities.
In graph theory, an st-planar graph is a bipolar orientation of a plane graph for which both the source and the sink of the orientation are on the outer face of the graph. That is, it is a directed graph drawn without crossings in the plane, in such a way that there are no directed cycles in the graph, exactly one graph vertex has no incoming edges, exactly one graph vertex has no outgoing edges, and these two special vertices both lie on the outer face of the graph.