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Graph drawing

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</style><div role="note" class="hatnote navigation-not-searchable">"Network diagram" redirects here. For network diagrams in specific contexts, see <a href="#Application-specific_graph_drawings">§ Application-specific graph drawings</a>.</div>

Graph drawing is an area of <a href="Mathematics" title="Mathematics">mathematics</a> and <a href="Computer_science" title="Computer science">computer science</a> combining methods from <a href="Geometric_graph_theory" title="Geometric graph theory">geometric graph theory</a> and <a href="Information_visualization" class="mw-redirect" title="Information visualization">information visualization</a> to derive two-dimensional (or, sometimes, three-dimensional) depictions of <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a> arising from applications such as <a href="Social_network_analysis" title="Social network analysis">social network analysis</a>, <a href="Cartography" title="Cartography">cartography</a>, <a href="Linguistics" title="Linguistics">linguistics</a>, and <a href="Bioinformatics" title="Bioinformatics">bioinformatics</a>.<a href="#cite_note-1">[1]</a>

A drawing of a graph or network diagram is a pictorial representation of the <a href="Vertex_(graph_theory)" title="Vertex (graph theory)">vertices</a> and <a href="Edge_(graph_theory)" class="mw-redirect" title="Edge (graph theory)">edges</a> of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph.<a href="#cite_note-dett6-2">[2]</a> 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 <a href="Aesthetics" title="Aesthetics">aesthetics</a>.<a href="#cite_note-dett-viii-3">[3]</a> 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.<a href="#cite_note-FOOTNOTEMisueEadesLaiSugiyama1995-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="Graphical_conventions">Graphical conventions</h2></div>

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 <a href="Line_segment" title="Line segment">line segments</a>, <a href="Polygonal_chain" title="Polygonal chain">polylines</a>, or curves in the <a href="Euclidean_plane" title="Euclidean plane">Euclidean plane</a>.<a href="#cite_note-dett-viii-3">[3]</a> Node–link diagrams can be traced back to the 14th-16th century works of Pseudo-Lull which were published under the name of <a href="Ramon_Llull" title="Ramon Llull">Ramon Llull</a>, a 13th century polymath. Pseudo-Lull drew diagrams of this type for <a href="Complete_graph" title="Complete graph">complete graphs</a> in order to analyze all pairwise combinations among sets of metaphysical concepts.<a href="#cite_note-FOOTNOTEKnuth2013-5">[5]</a>

In the case of <a href="Directed_graph" title="Directed graph">directed graphs</a>, <a href="Arrow_(symbol)" title="Arrow (symbol)">arrowheads</a> form a commonly used graphical convention to show their <a href="Orientability" title="Orientability">orientation</a>;<a href="#cite_note-dett6-2">[2]</a> however, user studies have shown that other conventions such as tapering provide this information more effectively.<a href="#cite_note-6">[6]</a> <a href="Upward_planar_drawing" title="Upward planar drawing">Upward planar drawing</a> uses the convention that every edge is oriented from a lower vertex to a higher vertex, making arrowheads unnecessary.<a href="#cite_note-FOOTNOTEGargTamassia1995-7">[7]</a>

Alternative conventions to node–link diagrams include adjacency representations such as <a href="Circle_packing_theorem" title="Circle packing theorem">circle packings</a>, in which vertices are represented by disjoint regions in the plane and edges are represented by adjacencies between regions; <a href="Intersection_graph" title="Intersection graph">intersection representations</a> in which vertices are represented by non-disjoint geometric objects and edges are represented by their intersections; <a href="Visibility_graph" title="Visibility graph">visibility representations</a> 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 <a href="Train_track_(mathematics)" title="Train track (mathematics)">train tracks</a>; fabrics, in which nodes are represented as horizontal lines and edges as vertical lines;<a href="#cite_note-Longabaugh_2012-8">[8]</a> and visualizations of the <a href="Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a> of the graph.

<div class="mw-heading mw-heading2"><h2 id="Quality_measures">Quality measures</h2></div>

Many different quality measures have been defined for graph drawings, in an attempt to find objective means of evaluating their aesthetics and usability.<a href="#cite_note-9">[9]</a> In addition to guiding the choice between different layout methods for the same graph, some layout methods attempt to directly optimize these measures.

<ul><li>The <a href="Crossing_number_(graph_theory)" title="Crossing number (graph theory)">crossing number</a> of a drawing is the number of pairs of edges that cross each other. If the graph is <a href="Planar_graph" title="Planar graph">planar</a>, then it is often convenient to draw it without any edge intersections; that is, in this case, a graph drawing represents a <a href="Graph_embedding" title="Graph embedding">graph embedding</a>. However, nonplanar graphs frequently arise in applications, so graph drawing algorithms must generally allow for edge crossings.<a href="#cite_note-10">[10]</a></li>

<li>The <a href="Area_(graph_drawing)" title="Area (graph drawing)">area</a> of a drawing is the size of its smallest <a href="Bounding_box" class="mw-redirect" title="Bounding box">bounding box</a>, relative to the closest distance between any two vertices. Drawings with smaller area are generally preferable to those with larger area, because they allow the features of the drawing to be shown at greater size and therefore more legibly. The <a href="Aspect_ratio" title="Aspect ratio">aspect ratio</a> of the bounding box may also be important.</li>

<li>Symmetry display is the problem of finding <a href="Graph_automorphism" title="Graph automorphism">symmetry groups</a> within a given graph, and finding a drawing that displays as much of the symmetry as possible. Some layout methods automatically lead to symmetric drawings; alternatively, some drawing methods start by finding symmetries in the input graph and using them to construct a drawing.<a href="#cite_note-11">[11]</a></li>

<li>It is important that edges have shapes that are as simple as possible, to make it easier for the eye to follow them. In polyline drawings, the complexity of an edge may be measured by its <a href="Bend_minimization" title="Bend minimization">number of bends</a>, and many methods aim to provide drawings with few total bends or few bends per edge. Similarly for spline curves the complexity of an edge may be measured by the number of control points on the edge.</li>

<li>Several commonly used quality measures concern lengths of edges: it is generally desirable to minimize the total length of the edges as well as the maximum length of any edge. Additionally, it may be preferable for the lengths of edges to be uniform rather than highly varied.</li>

<li><a href="Angular_resolution_(graph_drawing)" title="Angular resolution (graph drawing)">Angular resolution</a> is a measure of the sharpest angles in a graph drawing. If a graph has vertices with high <a href="Degree_(graph_theory)" title="Degree (graph theory)">degree</a> then it necessarily will have small angular resolution, but the angular resolution can be bounded below by a function of the degree.<a href="#cite_note-FOOTNOTEPachSharir2009-12">[12]</a></li>

<li>The <a href="Slope_number" title="Slope number">slope number</a> of a graph is the minimum number of distinct edge slopes needed in a drawing with straight line segment edges (allowing crossings). <a href="Cubic_graph" title="Cubic graph">Cubic graphs</a> have slope number at most four, but graphs of degree five may have unbounded slope number; it remains open whether the slope number of degree-4 graphs is bounded.<a href="#cite_note-FOOTNOTEPachSharir2009-12">[12]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Layout_methods">Layout methods</h2></div>

There are many different graph layout strategies:

<ul><li>In <a href="Force-based_layout" class="mw-redirect" title="Force-based layout">force-based layout</a> systems, the graph drawing software modifies an initial vertex placement by continuously moving the vertices according to a system of forces based on physical metaphors related to systems of <a href="Spring_(device)" title="Spring (device)">springs</a> or <a href="Molecular_mechanics" title="Molecular mechanics">molecular mechanics</a>. Typically, these systems combine attractive forces between adjacent vertices with repulsive forces between all pairs of vertices, in order to seek a layout in which edge lengths are small while vertices are well-separated. These systems may perform <a href="Gradient_descent" title="Gradient descent">gradient descent</a> based minimization of an <a href="Energy_function" class="mw-redirect" title="Energy function">energy function</a>, or they may translate the forces directly into velocities or accelerations for the moving vertices.<a href="#cite_note-14">[14]</a></li>

<li><a href="Spectral_layout" title="Spectral layout">Spectral layout</a> methods use as coordinates the <a href="Eigenvector" class="mw-redirect" title="Eigenvector">eigenvectors</a> of a <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> such as the <a href="Discrete_Laplace_operator" title="Discrete Laplace operator">Laplacian</a> derived from the <a href="Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a> of the graph.<a href="#cite_note-15">[15]</a></li>

<li>Orthogonal layout methods, which allow the edges of the graph to run horizontally or vertically, parallel to the coordinate axes of the layout. These methods were originally designed for <a href="VLSI" class="mw-redirect" title="VLSI">VLSI</a> and <a href="Printed_circuit_board" title="Printed circuit board">PCB</a> layout problems but they have also been adapted for graph drawing. They typically involve a multiphase approach in which an input graph is planarized by replacing crossing points by vertices, a topological embedding of the planarized graph is found, edge orientations are chosen to minimize bends, vertices are placed consistently with these orientations, and finally a layout compaction stage reduces the area of the drawing.<a href="#cite_note-16">[16]</a></li>

<li>Tree layout algorithms these show a rooted <a href="Tree_structure" title="Tree structure">tree</a>-like formation, suitable for <a href="Tree_(graph_theory)" title="Tree (graph theory)">trees</a>. Often, in a technique called "balloon layout", the children of each node in the tree are drawn on a circle surrounding the node, with the radii of these circles diminishing at lower levels in the tree so that these circles do not overlap.<a href="#cite_note-17">[17]</a></li>

<li><a href="Layered_graph_drawing" title="Layered graph drawing">Layered graph drawing</a> methods (often called Sugiyama-style drawing) are best suited for <a href="Directed_acyclic_graph" title="Directed acyclic graph">directed acyclic graphs</a> or graphs that are nearly acyclic, such as the graphs of dependencies between modules or functions in a software system. In these methods, the nodes of the graph are arranged into horizontal layers using methods such as the <a href="Coffman%E2%80%93Graham_algorithm" title="Coffman–Graham algorithm">Coffman–Graham algorithm</a>, in such a way that most edges go downwards from one layer to the next; after this step, the nodes within each layer are arranged in order to minimize crossings.<a href="#cite_note-18">[18]</a></li></ul>

<ul><li><a href="Arc_diagram" title="Arc diagram">Arc diagrams</a>, a layout style dating back to the 1960s,<a href="#cite_note-19">[19]</a> place vertices on a line; edges may be drawn as semicircles above or below the line, or as smooth curves linked together from multiple semicircles.</li>

<li><a href="Circular_layout" title="Circular layout">Circular layout</a> methods place the vertices of the graph on a circle, choosing carefully the ordering of the vertices around the circle to reduce crossings and place adjacent vertices close to each other. Edges may be drawn either as chords of the circle or as arcs inside or outside of the circle. In some cases, multiple circles may be used.<a href="#cite_note-20">[20]</a></li>

<li><a href="Dominance_drawing" title="Dominance drawing">Dominance drawing</a> places vertices in such a way that one vertex is upwards, rightwards, or both of another if and only if it is <a href="Reachability" title="Reachability">reachable</a> from the other vertex. In this way, the layout style makes the reachability relation of the graph visually apparent.<a href="#cite_note-21">[21]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Application-specific_graph_drawings">Application-specific graph drawings</h2></div>

Graphs and graph drawings arising in other areas of application include

<ul><li><a href="Sociogram" title="Sociogram">Sociograms</a>, drawings of a <a href="Social_network" title="Social network">social network</a>, as often offered by <a href="Social_network_analysis_software" title="Social network analysis software">social network analysis software</a><a href="#cite_note-22">[22]</a></li>

<li><a href="Hasse_diagram" title="Hasse diagram">Hasse diagrams</a>, a type of graph drawing specialized to <a href="Partial_order" class="mw-redirect" title="Partial order">partial orders</a><a href="#cite_note-23">[23]</a></li>

<li><a href="Dessin_d'enfant" title="Dessin d'enfant">Dessin d'enfants</a>, a type of graph drawing used in <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a><a href="#cite_note-FOOTNOTEZapponi2003-24">[24]</a></li>

<li><a href="State_diagram" title="State diagram">State diagrams</a>, graphical representations of <a href="Finite-state_machine" title="Finite-state machine">finite-state machines</a><a href="#cite_note-FOOTNOTEAndersonHead2006-25">[25]</a></li>

<li><a href="Computer_network_diagram" title="Computer network diagram">Computer network diagrams</a>, depictions of the nodes and connections in a <a href="Computer_network" title="Computer network">computer network</a><a href="#cite_note-FOOTNOTEDi_BattistaRimondini2014-26">[26]</a></li>

<li><a href="Flowchart" title="Flowchart">Flowcharts</a> and <a href="DRAKON" title="DRAKON">drakon-charts</a>, drawings in which the nodes represent the steps of an <a href="Algorithm" title="Algorithm">algorithm</a> and the edges represent <a href="Control_flow" title="Control flow">control flow</a> between steps.</li>

<li><a href="Project_network" title="Project network">Project network</a>, graphical depiction of the chronological order in which activities of a <a href="Project_management" title="Project management">project</a> are to be completed.</li>

<li><a href="Data-flow_diagram" title="Data-flow diagram">Data-flow diagrams</a>, drawings in which the nodes represent the components of an <a href="Information_system" title="Information system">information system</a> and the edges represent the movement of information from one component to another.</li>

<li><a href="Bioinformatics" title="Bioinformatics">Bioinformatics</a> including <a href="Phylogenetic_tree" title="Phylogenetic tree">phylogenetic trees</a>, <a href="Protein%E2%80%93protein_interaction" title="Protein–protein interaction">protein–protein interaction</a> networks, and <a href="Metabolic_pathway" title="Metabolic pathway">metabolic pathways</a>.<a href="#cite_note-FOOTNOTEBachmaierBrandesSchreiber2014-27">[27]</a></li></ul>

In addition, the <a href="Placement_(electronic_design_automation)" title="Placement (electronic design automation)">placement</a> and <a href="Routing_(electronic_design_automation)" title="Routing (electronic design automation)">routing</a> steps of <a href="Electronic_design_automation" title="Electronic design automation">electronic design automation</a> (EDA) are similar in many ways to graph drawing, as is the problem of <a href="Greedy_embedding" title="Greedy embedding">greedy embedding</a> in <a href="Distributed_computing" title="Distributed computing">distributed computing</a>, 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.

<div class="mw-heading mw-heading2"><h2 id="Graph_drawing_algorithms">Graph drawing algorithms</h2></div>

There are many algorithms for graph drawing. Among them are:

<ul><li>The Reingold-Tilford algorithm for tree drawing.<a href="#cite_note-FOOTNOTEReingoldTilford1981-28">[28]</a></li>

<li>Kant's algorithm,<a href="#cite_note-FOOTNOTEKant1992-29">[29]</a> which constructs a polyline drawing of a 3-connected planar graph such that the size of the minimal angle among arcs is at least <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{d}}\pi }">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{d}}\pi }</annotation>

</semantics>

</math><img src="./52843840f0191d64ef32a9e5c52f1b2ce8a30f70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.384ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{d}}\pi }" loading="lazy">, where d is the maximal node degree; and its generaliztion, that also works well for other planar graphs, by Gutwenger and Mutzel.<a href="#cite_note-FOOTNOTEGutwengerMutzel1998-30">[30]</a></li>

<li>Tamassia's algorithm for minimizing the number of bends in an orthogonal representation of a planar graph.<a href="#cite_note-FOOTNOTETamassia1987-31">[31]</a></li>

<li>The Magnetic Spring Model by Sugiyama and Misue.<a href="#cite_note-FOOTNOTESugiyamaMisue1995-32">[32]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Software">Software</h2></div>

Software, systems, and providers of systems for drawing graphs include:

<ul><li><a href="BioFabric" title="BioFabric">BioFabric</a> open-source software for visualizing large networks by drawing nodes as horizontal lines.</li>

<li><a href="Cytoscape" title="Cytoscape">Cytoscape</a>, open-source software for visualizing molecular interaction networks</li>

<li><a href="Gephi" title="Gephi">Gephi</a>, open-source network analysis and visualization software</li>

<li><a href="Graph-tool" title="Graph-tool">graph-tool</a>, a <a href="Free_Software" class="mw-redirect" title="Free Software">free/libre</a> <a href="Python_(programming_language)" title="Python (programming language)">Python</a> library for analysis of graphs</li>

<li><a href="Graphviz" title="Graphviz">Graphviz</a>, an open-source graph drawing system from <a href="AT%26T_Corporation" title="AT&T Corporation">AT&T Corporation</a><a href="#cite_note-33">[33]</a></li>

<li><a href="Linkurious" title="Linkurious">Linkurious</a>, a commercial network analysis and visualization software for <a href="Graph_databases" class="mw-redirect" title="Graph databases">graph databases</a></li>

<li><a href="Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a>, a general-purpose computation tool that includes 2D and 3D graph visualization and graph analysis tools.<a href="#cite_note-34">[34]</a></li>

<li><a href="Microsoft_Automatic_Graph_Layout" title="Microsoft Automatic Graph Layout">Microsoft Automatic Graph Layout</a>, open-source .NET library (formerly called GLEE) for laying out graphs<a href="#cite_note-35">[35]</a></li>

<li><a href="NetworkX" title="NetworkX">NetworkX</a> is a Python library for studying graphs and networks.</li>

<li><a href="Tulip_(software)" title="Tulip (software)">Tulip</a>,<a href="#cite_note-36">[36]</a> an open-source data visualization tool</li>

<li><a href="YEd" title="YEd">yEd</a>, a graph editor with graph layout functionality<a href="#cite_note-37">[37]</a></li>

<li><a href="PGF/TikZ" title="PGF/TikZ">PGF/TikZ</a> 3.0 with the <code>graphdrawing</code> package (requires <a href="LuaTeX" title="LuaTeX">LuaTeX</a>).<a href="#cite_note-38">[38]</a></li>

<li><a href="LaNet-vi" title="LaNet-vi">LaNet-vi</a>, an open-source large network visualization software</li>

<li><a href="Open_Graph_Drawing_Framework" title="Open Graph Drawing Framework">OGDF</a>, an open-source library of C++ data structures and algorithms, mostly for graph drawing</li></ul>

<ul><li><a href="International_Symposium_on_Graph_Drawing" title="International Symposium on Graph Drawing">International Symposium on Graph Drawing</a></li>

<li><a href="List_of_Unified_Modeling_Language_tools" title="List of Unified Modeling Language tools">List of Unified Modeling Language tools</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Footnotes">Footnotes</h3></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="#CITEREFDi_BattistaEadesTamassiaTollis1998">Di Battista et al. (1998)</a>, pp. vii–viii; <a href="#CITEREFHermanMelançonMarshall2000">Herman, Melançon & Marshall (2000)</a>, Section 1.1, "Typical Application Areas".

</li>

<li id="cite_note-dett6-2">^ <a href="#cite_ref-dett6_2-0">a</a> <a href="#cite_ref-dett6_2-1">b</a> <a href="#CITEREFDi_BattistaEadesTamassiaTollis1998">Di Battista et al. (1998)</a>, p. 6.

</li>

<li id="cite_note-dett-viii-3">^ <a href="#cite_ref-dett-viii_3-0">a</a> <a href="#cite_ref-dett-viii_3-1">b</a> <a href="#CITEREFDi_BattistaEadesTamassiaTollis1998">Di Battista et al. (1998)</a>, p. viii.

</li>

<li id="cite_note-FOOTNOTEMisueEadesLaiSugiyama1995-4"><a href="#cite_ref-FOOTNOTEMisueEadesLaiSugiyama1995_4-0">^</a> <a href="#CITEREFMisueEadesLaiSugiyama1995">Misue et al. (1995)</a>.

</li>

<li id="cite_note-FOOTNOTEKnuth2013-5"><a href="#cite_ref-FOOTNOTEKnuth2013_5-0">^</a> <a href="#CITEREFKnuth2013">Knuth (2013)</a>.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="#CITEREFHoltenvan_Wijk2009">Holten & van Wijk (2009)</a>; <a href="#CITEREFHoltenIsenbergvan_WijkFekete2011">Holten et al. (2011)</a>.

</li>

<li id="cite_note-FOOTNOTEGargTamassia1995-7"><a href="#cite_ref-FOOTNOTEGargTamassia1995_7-0">^</a> <a href="#CITEREFGargTamassia1995">Garg & Tamassia (1995)</a>.

</li>

<li id="cite_note-Longabaugh_2012-8"><a href="#cite_ref-Longabaugh_2012_8-0">^</a> <a href="#CITEREFLongabaugh2012">Longabaugh (2012)</a>.

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <a href="#CITEREFDi_BattistaEadesTamassiaTollis1998">Di Battista et al. (1998)</a>, Section 2.1.2, Aesthetics, pp. 14–16; <a href="#CITEREFPurchaseCohenJames1997">Purchase, Cohen & James (1997)</a>.

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <a href="#CITEREFDi_BattistaEadesTamassiaTollis1998">Di Battista et al. (1998)</a>, p 14.

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <a href="#CITEREFDi_BattistaEadesTamassiaTollis1998">Di Battista et al. (1998)</a>, p. 16.

</li>

<li id="cite_note-FOOTNOTEPachSharir2009-12">^ <a href="#cite_ref-FOOTNOTEPachSharir2009_12-0">a</a> <a href="#cite_ref-FOOTNOTEPachSharir2009_12-1">b</a> <a href="#CITEREFPachSharir2009">Pach & Sharir (2009)</a>.

</li>

<li id="cite_note-FOOTNOTEGrandjean2014-13"><a href="#cite_ref-FOOTNOTEGrandjean2014_13-0">^</a> <a href="#CITEREFGrandjean2014">Grandjean (2014)</a>.

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <a href="#CITEREFDi_BattistaEadesTamassiaTollis1998">Di Battista et al. (1998)</a>, Section 2.7, "The Force-Directed Approach", pp. 29–30, and Chapter 10, "Force-Directed Methods", pp. 303–326.

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> <a href="#CITEREFBeckman1994">Beckman (1994)</a>; <a href="#CITEREFKoren2005">Koren (2005)</a>.

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <a href="#CITEREFDi_BattistaEadesTamassiaTollis1998">Di Battista et al. (1998)</a>, Chapter 5, "Flow and Orthogonal Drawings", pp. 137–170; <a href="#CITEREFEiglspergerFeketeKlau2001">Eiglsperger, Fekete & Klau (2001)</a>.

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <a href="#CITEREFHermanMelançonMarshall2000">Herman, Melançon & Marshall (2000)</a>, Section 2.2, "Traditional Layout – An Overview".

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <a href="#CITEREFSugiyamaTagawaToda1981">Sugiyama, Tagawa & Toda (1981)</a>; <a href="#CITEREFBastertMatuszewski2001">Bastert & Matuszewski (2001)</a>; <a href="#CITEREFDi_BattistaEadesTamassiaTollis1998">Di Battista et al. (1998)</a>, Chapter 9, "Layered Drawings of Digraphs", pp. 265–302.

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> <a href="#CITEREFSaaty1964">Saaty (1964)</a>.

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> <a href="#CITEREFDoğrusözMaddenMadden1997">Doğrusöz, Madden & Madden (1997)</a>.

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> <a href="#CITEREFDi_BattistaEadesTamassiaTollis1998">Di Battista et al. (1998)</a>, Section 4.7, "Dominance Drawings", pp. 112–127.

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> <a href="#CITEREFScott2000">Scott (2000)</a>; <a href="#CITEREFBrandesFreemanWagner2014">Brandes, Freeman & Wagner (2014)</a>.

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> <a href="#CITEREFDi_BattistaEadesTamassiaTollis1998">Di Battista et al. (1998)</a>, pp. 15–16, and Chapter 6, "Flow and Upward Planarity", pp. 171–214; <a href="#CITEREFFreese2004">Freese (2004)</a>.

</li>

<li id="cite_note-FOOTNOTEZapponi2003-24"><a href="#cite_ref-FOOTNOTEZapponi2003_24-0">^</a> <a href="#CITEREFZapponi2003">Zapponi (2003)</a>.

</li>

<li id="cite_note-FOOTNOTEAndersonHead2006-25"><a href="#cite_ref-FOOTNOTEAndersonHead2006_25-0">^</a> <a href="#CITEREFAndersonHead2006">Anderson & Head (2006)</a>.

</li>

<li id="cite_note-FOOTNOTEDi_BattistaRimondini2014-26"><a href="#cite_ref-FOOTNOTEDi_BattistaRimondini2014_26-0">^</a> <a href="#CITEREFDi_BattistaRimondini2014">Di Battista & Rimondini (2014)</a>.

</li>

<li id="cite_note-FOOTNOTEBachmaierBrandesSchreiber2014-27"><a href="#cite_ref-FOOTNOTEBachmaierBrandesSchreiber2014_27-0">^</a> <a href="#CITEREFBachmaierBrandesSchreiber2014">Bachmaier, Brandes & Schreiber (2014)</a>.

</li>

<li id="cite_note-FOOTNOTEReingoldTilford1981-28"><a href="#cite_ref-FOOTNOTEReingoldTilford1981_28-0">^</a> <a href="#CITEREFReingoldTilford1981">Reingold & Tilford (1981)</a>.

</li>

<li id="cite_note-FOOTNOTEKant1992-29"><a href="#cite_ref-FOOTNOTEKant1992_29-0">^</a> <a href="#CITEREFKant1992">Kant (1992)</a>.

</li>

<li id="cite_note-FOOTNOTEGutwengerMutzel1998-30"><a href="#cite_ref-FOOTNOTEGutwengerMutzel1998_30-0">^</a> <a href="#CITEREFGutwengerMutzel1998">Gutwenger & Mutzel (1998)</a>.

</li>

<li id="cite_note-FOOTNOTETamassia1987-31"><a href="#cite_ref-FOOTNOTETamassia1987_31-0">^</a> <a href="#CITEREFTamassia1987">Tamassia (1987)</a>.

</li>

<li id="cite_note-FOOTNOTESugiyamaMisue1995-32"><a href="#cite_ref-FOOTNOTESugiyamaMisue1995_32-0">^</a> <a href="#CITEREFSugiyamaMisue1995">Sugiyama & Misue (1995)</a>.

</li>

<li id="cite_note-33"><a href="#cite_ref-33">^</a> "Graphviz and Dynagraph – Static and Dynamic Graph Drawing Tools", by John Ellson, Emden R. Gansner, Eleftherios Koutsofios, Stephen C. North, and Gordon Woodhull, in <a href="#CITEREFJüngerMutzel2004">Jünger & Mutzel (2004)</a>.

</li>

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</style><cite class="citation cs2"><a rel="nofollow" class="external text" href="http://reference.wolfram.com/mathematica/tutorial/GraphDrawingIntroduction.html">"Introduction to graph drawing"</a>, Wolfram Language & System Documentation Center, retrieved 2024-03-21</cite>

</li>

<li id="cite_note-35"><a href="#cite_ref-35">^</a> <a href="#CITEREFNachmansonRobertsonLee2008">Nachmanson, Robertson & Lee (2008)</a>.

</li>

<li id="cite_note-36"><a href="#cite_ref-36">^</a> "Tulip – A Huge Graph Visualization Framework", by David Auber, in <a href="#CITEREFJüngerMutzel2004">Jünger & Mutzel (2004)</a>.

</li>

<li id="cite_note-37"><a href="#cite_ref-37">^</a> "yFiles – Visualization and Automatic Layout of Graphs", by Roland Wiese, Markus Eiglsperger, and Michael Kaufmann, in <a href="#CITEREFJüngerMutzel2004">Jünger & Mutzel (2004)</a>.

</li>

<li id="cite_note-38"><a href="#cite_ref-38">^</a> <a href="#CITEREFTantau2013">Tantau (2013)</a>; see also the older <a rel="nofollow" class="external text" href="http://www.tcs.uni-luebeck.de/downloads/mitarbeiter/tantau/2012-gd-presentation.pdf">GD 2012 presentation</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160527034523/http://www.tcs.uni-luebeck.de/downloads/mitarbeiter/tantau/2012-gd-presentation.pdf">Archived</a> 2016-05-27 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>

</li>

</ol></div>

<div class="mw-heading mw-heading3"><h3 id="General_references">General references</h3></div>

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</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em">

<ul><li><cite id="CITEREFDi_BattistaEadesTamassiaTollis1998" class="citation cs2">Di Battista, Giuseppe; <a href="Peter_Eades" title="Peter Eades">Eades, Peter</a>; <a href="Roberto_Tamassia" title="Roberto Tamassia">Tamassia, Roberto</a>; Tollis, Ioannis G. (1998), Graph Drawing: Algorithms for the Visualization of Graphs, <a href="Prentice_Hall" title="Prentice Hall">Prentice Hall</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-301615-4</bdi></cite>.</li>

<li><cite id="CITEREFHermanMelançonMarshall2000" class="citation cs2">Herman, Ivan; Melançon, Guy; Marshall, M. Scott (2000), "Graph Visualization and Navigation in Information Visualization: A Survey", IEEE Transactions on Visualization and Computer Graphics, 6 (1): 24–43, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F2945.841119">10.1109/2945.841119</a></cite>.</li>

<li><cite id="CITEREFJüngerMutzel2004" class="citation cs2">Jünger, Michael; <a href="Petra_Mutzel" title="Petra Mutzel">Mutzel, Petra</a> (2004), Graph Drawing Software, Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-00881-1</bdi></cite>.</li></ul>

</div>

<div class="mw-heading mw-heading3"><h3 id="Specialized_subtopics">Specialized subtopics</h3></div>

<ul><li><cite id="CITEREFAndersonHead2006" class="citation cs2">Anderson, James Andrew; Head, Thomas J. (2006), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ikS8BLdLDxIC&pg=PA38">Automata Theory with Modern Applications</a>, Cambridge University Press, pp. 38–41, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-84887-9</bdi></cite>.</li>

<li><cite id="CITEREFBachmaierBrandesSchreiber2014" class="citation cs2">Bachmaier, Christian; <a href="Ulrik_Brandes" title="Ulrik Brandes">Brandes, Ulrik</a>; Schreiber, Falk (2014), "Biological Networks", in <a href="Roberto_Tamassia" title="Roberto Tamassia">Tamassia, Roberto</a> (ed.), Handbook of Graph Drawing and Visualization, CRC Press, pp. 621–651</cite>.</li>

<li><cite id="CITEREFBastertMatuszewski2001" class="citation cs2">Bastert, Oliver; Matuszewski, Christian (2001), "Layered drawings of digraphs", in Kaufmann, Michael; <a href="Dorothea_Wagner" title="Dorothea Wagner">Wagner, Dorothea</a> (eds.), Drawing Graphs: Methods and Models, Lecture Notes in Computer Science, vol. 2025, Springer-Verlag, pp. 87–120, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-44969-8_5">10.1007/3-540-44969-8_5</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-42062-0</bdi></cite>.</li>

<li><cite id="CITEREFBeckman1994" class="citation cs2">Beckman, Brian (1994), <a rel="nofollow" class="external text" href="http://research.microsoft.com/apps/pubs/default.aspx?id=69611">Theory of Spectral Graph Layout</a>, Tech. Report MSR-TR-94-04, Microsoft Research, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160401054322/http://research.microsoft.com/apps/pubs/default.aspx?id=69611">archived</a> from the original on 2016-04-01, retrieved 2011-09-17</cite>.</li>

<li><cite id="CITEREFBrandesFreemanWagner2014" class="citation cs2"><a href="Ulrik_Brandes" title="Ulrik Brandes">Brandes, Ulrik</a>; Freeman, Linton C.; <a href="Dorothea_Wagner" title="Dorothea Wagner">Wagner, Dorothea</a> (2014), "Social Networks", in <a href="Roberto_Tamassia" title="Roberto Tamassia">Tamassia, Roberto</a> (ed.), Handbook of Graph Drawing and Visualization, CRC Press, pp. 805–839</cite>.</li>

<li><cite id="CITEREFDi_BattistaRimondini2014" class="citation cs2">Di Battista, Giuseppe; Rimondini, Massimo (2014), "Computer Networks", in <a href="Roberto_Tamassia" title="Roberto Tamassia">Tamassia, Roberto</a> (ed.), Handbook of Graph Drawing and Visualization, CRC Press, pp. 763–803</cite>.</li>

<li><cite id="CITEREFDoğrusözMaddenMadden1997" class="citation cs2">Doğrusöz, Uğur; Madden, Brendan; Madden, Patrick (1997), "Circular layout in the Graph Layout toolkit", in North, Stephen (ed.), <a href="International_Symposium_on_Graph_Drawing" title="International Symposium on Graph Drawing">Symposium on Graph Drawing, GD '96 Berkeley, California, USA, September 18–20, 1996, Proceedings</a>, Lecture Notes in Computer Science, vol. 1190, Springer-Verlag, pp. 92–100, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-62495-3_40">10.1007/3-540-62495-3_40</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-62495-0</bdi></cite>.</li>

<li><cite id="CITEREFEiglspergerFeketeKlau2001" class="citation cs2">Eiglsperger, Markus; Fekete, Sándor; Klau, Gunnar (2001), "Orthogonal graph drawing", in Kaufmann, Michael; <a href="Dorothea_Wagner" title="Dorothea Wagner">Wagner, Dorothea</a> (eds.), Drawing Graphs, Lecture Notes in Computer Science, vol. 2025, Springer Berlin / Heidelberg, pp. 121–171, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-44969-8_6">10.1007/3-540-44969-8_6</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-42062-0</bdi></cite>.</li>

<li><cite id="CITEREFFreese2004" class="citation cs2">Freese, Ralph (2004), "Automated lattice drawing", in Eklund, Peter (ed.), <a rel="nofollow" class="external text" href="http://www.math.hawaii.edu/~ralph/Preprints/latdrawing.pdf">Concept Lattices: Second International Conference on Formal Concept Analysis, ICFCA 2004, Sydney, Australia, February 23-26, 2004, Proceedings</a> (PDF), Lecture Notes in Computer Science, vol. 2961, Springer-Verlag, pp. 589–590, <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.69.6245">10.1.1.69.6245</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-24651-0_12">10.1007/978-3-540-24651-0_12</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-21043-6</bdi>, <a rel="nofollow" class="external text" href="http://arquivo.pt/wayback/20160314184411/http://www.math.hawaii.edu/~ralph/Preprints/latdrawing.pdf">archived</a> (PDF) from the original on 2016-03-14, retrieved 2011-09-17</cite>.</li>

<li><cite id="CITEREFGargTamassia1995" class="citation cs2">Garg, Ashim; Tamassia, Roberto (1995), "Upward planarity testing", <a href="Order_(journal)" title="Order (journal)">Order</a>, 12 (2): 109–133, <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.10.2237">10.1.1.10.2237</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01108622">10.1007/BF01108622</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1354797">1354797</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14183717">14183717</a></cite>.</li>

<li><cite id="CITEREFGrandjean2014" class="citation cs2">Grandjean, Martin (2014), <a rel="nofollow" class="external text" href="http://www.cairn.info/resume.php?ID_ARTICLE=LCN_103_0037">"La connaissance est un réseau"</a>, Les Cahiers du Numérique, 10 (3): 37–54, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3166%2Flcn.10.3.37-54">10.3166/lcn.10.3.37-54</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150627140457/http://www.cairn.info/resume.php?ID_ARTICLE=LCN_103_0037">archived</a> from the original on 2015-06-27, retrieved 2014-10-15</cite>.</li>

<li><cite id="CITEREFGutwengerMutzel1998" class="citation conference cs1">Gutwenger, Carsten; Mutzel, Petra (1998). <a rel="nofollow" class="external text" href="https://doi.org/10.1007/3-540-37623-2_13">"Planar Polyline Drawings with Good Angular Resolution"</a>. In Sue Whitesides (ed.). Graph Drawing, 6th International Symposium. Springer. pp. 167–182. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-37623-2_13">10.1007/3-540-37623-2_13</a>.</cite></li>

<li><cite id="CITEREFHoltenIsenbergvan_WijkFekete2011" class="citation cs2">Holten, Danny; <a href="Petra_Isenberg" title="Petra Isenberg">Isenberg, Petra</a>; <a href="Jack_van_Wijk" title="Jack van Wijk">van Wijk, Jarke J.</a>; Fekete, Jean-Daniel (2011), "An extended evaluation of the readability of tapered, animated, and textured directed-edge representations in node-link graphs", <a rel="nofollow" class="external text" href="http://www.lri.fr/~isenberg/publications/papers/Holten_2011_AEP.pdf">IEEE Pacific Visualization Symposium (PacificVis 2011)</a> (PDF), pp. 195–202, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FPACIFICVIS.2011.5742390">10.1109/PACIFICVIS.2011.5742390</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-61284-935-5</bdi>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16526781">16526781</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160411130015/https://www.lri.fr/~isenberg/publications/papers/Holten_2011_AEP.pdf">archived</a> (PDF) from the original on 2016-04-11, retrieved 2011-09-29</cite>.</li>

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<li><cite id="CITEREFKant1992" class="citation conference cs1">Kant, Goos (1992). "Drawing planar graphs using the lmc-ordering". 33rd Annual Symposium on Foundations of Computer Science. IEEE. pp. 101–110.</cite></li>

<li><cite id="CITEREFKnuth2013" class="citation cs2"><a href="Donald_Knuth" title="Donald Knuth">Knuth, Donald E.</a> (2013), "Two thousand years of combinatorics", in Wilson, Robin; Watkins, John J. (eds.), Combinatorics: Ancient and Modern, Oxford University Press, pp. 7–37</cite>.</li>

<li><cite id="CITEREFKoren2005" class="citation cs2">Koren, Yehuda (2005), "Drawing graphs by eigenvectors: theory and practice", Computers & Mathematics with Applications, 49 (11–12): 1867–1888, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.camwa.2004.08.015">10.1016/j.camwa.2004.08.015</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2154691">2154691</a></cite>.</li>

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<li><cite id="CITEREFMaddenMaddenPowersHimsolt1996" class="citation cs2">Madden, Brendan; Madden, Patrick; Powers, Steve; Himsolt, Michael (1996), "Portable graph layout and editing", in Brandenburg, Franz J. (ed.), <a href="International_Symposium_on_Graph_Drawing" title="International Symposium on Graph Drawing">Graph Drawing: Symposium on Graph Drawing, GD '95, Passau, Germany, September 20–22, 1995, Proceedings</a>, Lecture Notes in Computer Science, vol. 1027, Springer-Verlag, pp. 385–395, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0021822">10.1007/BFb0021822</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-60723-6</bdi></cite>.</li>

<li><cite id="CITEREFMisueEadesLaiSugiyama1995" class="citation cs2">Misue, K.; Eades, P.; Lai, W.; Sugiyama, K. (1995), "Layout Adjustment and the Mental Map", Journal of Visual Languages & Computing, 6 (2): 183–210, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjvlc.1995.1010">10.1006/jvlc.1995.1010</a></cite>.</li>

<li><cite id="CITEREFNachmansonRobertsonLee2008" class="citation cs2">Nachmanson, Lev; Robertson, George; <a href="Bongshin_Lee" title="Bongshin Lee">Lee, Bongshin</a> (2008), "Drawing Graphs with GLEE", in <a href="Seok-Hee_Hong" title="Seok-Hee Hong">Hong, Seok-Hee</a>; <a href="Takao_Nishizeki" title="Takao Nishizeki">Nishizeki, Takao</a>; Quan, Wu (eds.), <a href="International_Symposium_on_Graph_Drawing" title="International Symposium on Graph Drawing">Graph Drawing, 15th International Symposium, GD 2007, Sydney, Australia, September 24–26, 2007, Revised Papers</a>, Lecture Notes in Computer Science, vol. 4875, Springer-Verlag, pp. 389–394, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-77537-9_38">10.1007/978-3-540-77537-9_38</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-77536-2</bdi></cite>.</li>

<li><cite id="CITEREFPachSharir2009" class="citation cs2"><a href="J%C3%A1nos_Pach" title="János Pach">Pach, János</a>; <a href="Micha_Sharir" title="Micha Sharir">Sharir, Micha</a> (2009), "5.5 Angular resolution and slopes", Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures, Mathematical Surveys and Monographs, vol. 152, <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, pp. 126–127</cite>.</li>

<li><cite id="CITEREFPurchaseCohenJames1997" class="citation cs2"><a href="Helen_Purchase" title="Helen Purchase">Purchase, H. C.</a>; Cohen, R. F.; James, M. I. (1997), "An experimental study of the basis for graph drawing algorithms", Journal of Experimental Algorithmics, 2, Article 4, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F264216.264222">10.1145/264216.264222</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:22076200">22076200</a></cite>.</li>

<li><cite id="CITEREFReingoldTilford1981" class="citation journal cs1">Reingold, Edward M.; Tilford, John S. (1981). "Tidier drawings of trees". IEEE Transactions on Software Engineering. 2 (2): 223–228. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTSE.1981.234519">10.1109/TSE.1981.234519</a>.</cite></li>

<li><cite id="CITEREFSaaty1964" class="citation cs2"><a href="Thomas_L._Saaty" title="Thomas L. Saaty">Saaty, Thomas L.</a> (1964), "The minimum number of intersections in complete graphs", Proc. Natl. Acad. Sci. U.S.A., 52 (3): 688–690, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1964PNAS...52..688S">1964PNAS...52..688S</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.52.3.688">10.1073/pnas.52.3.688</a>, <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC300329">300329</a>, <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16591215">16591215</a></cite>.</li>

<li><cite id="CITEREFScott2000" class="citation cs2">Scott, John (2000), "Sociograms and Graph Theory", <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ww3_bKcz6kgC&pg=PA">Social network analysis: a handbook</a> (2nd ed.), Sage, pp. 64–69, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7619-6339-4</bdi></cite>.</li>

<li><cite id="CITEREFSugiyamaTagawaToda1981" class="citation cs2 cs1-prop-long-vol"><a href="Kozo_Sugiyama" title="Kozo Sugiyama">Sugiyama, Kozo</a>; Tagawa, Shôjirô; Toda, Mitsuhiko (1981), "Methods for visual understanding of hierarchical system structures", <a href="IEEE_Systems%2C_Man%2C_and_Cybernetics_Society" title="IEEE Systems, Man, and Cybernetics Society">IEEE Transactions on Systems, Man, and Cybernetics</a>, SMC-11 (2): 109–125, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTSMC.1981.4308636">10.1109/TSMC.1981.4308636</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0611436">0611436</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:8367756">8367756</a></cite>.</li>

<li><cite id="CITEREFSugiyamaMisue1995" class="citation journal cs1">Sugiyama, Kozo; Misue, Kazuo (1995). "Graph drawing by the magnetic spring model". Journal of Visual Languages & Computing. 6 (3): 217–231. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjvlc.1995.1013">10.1006/jvlc.1995.1013</a>.</cite></li>

<li><cite id="CITEREFTamassia1987" class="citation journal cs1">Tamassia, Roberto (1987). <a rel="nofollow" class="external text" href="https://doi.org/10.1137/0216030">"On Embedding a Graph in the Grid with the Minimum Number of Bends"</a>. SIAM Journal on Computing. 16 (3): 421–444. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F0216030">10.1137/0216030</a>.</cite></li>

<li><cite id="CITEREFTantau2013" class="citation cs2">Tantau, Till (2013), "Graph Drawing in TikZ", <a href="Journal_of_Graph_Algorithms_and_Applications" title="Journal of Graph Algorithms and Applications">Journal of Graph Algorithms and Applications</a>, 17 (4): 495–513, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.7155%2Fjgaa.00301">10.7155/jgaa.00301</a> (inactive July 15, 2025)</cite><code class="cs1-code">{{citation}}</code>: CS1 maint: DOI inactive as of July 2025 (link).</li>

<li><cite id="CITEREFZapponi2003" class="citation cs2">Zapponi, Leonardo (August 2003), <a rel="nofollow" class="external text" href="https://www.ams.org/notices/200307/what-is.pdf">"What is a Dessin d'Enfant"</a> (PDF), <a href="Notices_of_the_American_Mathematical_Society" title="Notices of the American Mathematical Society">Notices of the American Mathematical Society</a>, 50: 788–789, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211003075615/https://www.ams.org/notices/200307/what-is.pdf">archived</a> (PDF) from the original on 2021-10-03, retrieved 2021-04-28</cite>.</li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFDi_BattistaEadesTamassiaTollis1994" class="citation cs2">Di Battista, Giuseppe; <a href="Peter_Eades" title="Peter Eades">Eades, Peter</a>; <a href="Roberto_Tamassia" title="Roberto Tamassia">Tamassia, Roberto</a>; Tollis, Ioannis G. (1994), "Algorithms for Drawing Graphs: an Annotated Bibliography", <a href="Computational_Geometry_(journal)" title="Computational Geometry (journal)">Computational Geometry: Theory and Applications</a>, 4 (5): 235–282, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0925-7721%2894%2900014-x">10.1016/0925-7721(94)00014-x</a></cite>.</li>

<li><cite id="CITEREFKaufmannWagner2001" class="citation cs2">Kaufmann, Michael; <a href="Dorothea_Wagner" title="Dorothea Wagner">Wagner, Dorothea</a>, eds. (2001), Drawing Graphs: Methods and Models, <a href="Lecture_Notes_in_Computer_Science" title="Lecture Notes in Computer Science">Lecture Notes in Computer Science</a>, vol. 2025, Springer-Verlag, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-44969-8">10.1007/3-540-44969-8</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-42062-0</bdi>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1808286">1808286</a></cite>.</li>

<li><cite id="CITEREFTamassia2014" class="citation cs2"><a href="Roberto_Tamassia" title="Roberto Tamassia">Tamassia, Roberto</a>, ed. (2014), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130815181243/http://cs.brown.edu/~rt/gdhandbook/">Handbook of Graph Drawing and Visualization</a>, CRC Press, archived from <a rel="nofollow" class="external text" href="http://cs.brown.edu/~rt/gdhandbook/">the original</a> on 2013-08-15, retrieved 2013-08-28</cite>.</li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://graphx.codeplex.com">GraphX library for .NET</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180126071742/http://graphx.codeplex.com/">Archived</a> 2018-01-26 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>: open-source WPF library for graph calculation and visualization. Supports many layout and edge routing algorithms.</li>

<li><a rel="nofollow" class="external text" href="http://gdea.informatik.uni-koeln.de/">Graph drawing e-print archive</a>: including information on papers from all <a href="International_Symposium_on_Graph_Drawing" title="International Symposium on Graph Drawing">Graph Drawing symposia</a>.</li></ul>

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</style><div id="Graph_representations87" style="font-size:114%;margin:0 4em">Graph representations</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Data structures</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Graph_(abstract_data_type)" title="Graph (abstract data type)">Graph (abstract data type)</a></li>

<li><a href="Adjacency_list" title="Adjacency list">Adjacency list</a></li>

<li><a href="Adjacency_matrix" title="Adjacency matrix">Adjacency matrix</a></li>

<li><a href="Incidence_matrix" title="Incidence matrix">Incidence matrix</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">XML-based formats</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li>DotML</li>

<li><a href="GraphML" title="GraphML">GraphML</a></li>

<li><a href="XGMML" title="XGMML">XGMML</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Text-based formats</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Graph_Modelling_Language" title="Graph Modelling Language">Graph Modelling Language</a> (GML)</li>

<li><a href="LCF_notation" title="LCF notation">LCF notation</a> for cubic Hamiltonian graphs</li>

<li><a href="Newick_format" title="Newick format">Newick format</a> for trees</li>

<li><a href="Trivial_Graph_Format" title="Trivial Graph Format">Trivial Graph Format</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related concepts</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Graph_database" title="Graph database">Graph database</a></li>

<li><a href="Linked_data" title="Linked data">Linked data</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-15" href="https://en.wikipedia.org/wiki/?title=Graph_drawing&oldid=1300580473">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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