Arc diagram

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.

An arc diagram of the Goldner–Harary graph. This graph is not Hamiltonian, but can be made Hamiltonian by subdividing the edge crossed by the red dashed line segment and adding two edges along this segment.

The use of the phrase "arc diagram" for this kind of drawing follows the use of a similar type of diagram by Wattenberg (2002) to visualize the repetition patterns in strings, by using arcs to connect pairs of equal substrings. However, this style of graph drawing is much older than its name, dating back to the work of Saaty (1964) and Nicholson (1968), who used arc diagrams to study crossing numbers of graphs. An older but less frequently used name for arc diagrams is linear embeddings. More recently, arc diagrams have been used within the framework of circuit topology of knots and tangles, where they are termed as circuit diagrams.

Heer, Bostock & Ogievetsky (2010) write that arc diagrams "may not convey the overall structure of the graph as effectively as a two-dimensional layout", but that their layout makes it easy to display multivariate data associated with the vertices of the graph. Applications of arc diagrams include the Farey diagram, a visualization of number-theoretic connections between rational numbers, and diagrams representing RNA secondary structure in which the crossings of the diagram represent pseudoknots in the structure.

Planar graphs

As Nicholson (1968) observed, every drawing of a graph in the plane may be deformed into an arc diagram, without changing its number of crossings. In particular, every planar graph has a planar arc diagram. However, this embedding may need to use more than one semicircle for some of its edges.

If a graph is drawn without crossings using an arc diagram in which each edge is a single semicircle, then the drawing is a two-page book embedding, something that is only possible for the subhamiltonian graphs, a proper subset of the planar graphs. For instance, a maximal planar graph has such an embedding if and only if it contains a Hamiltonian cycle. Therefore, a non-Hamiltonian maximal planar graph such as the Goldner–Harary graph cannot have a planar embedding with one semicircle per edge. Testing whether a given graph has a crossing-free arc diagram of this type (or equivalently, whether it has pagenumber two) is NP-complete.

However, every planar graph has an arc diagram in which each edge is drawn as a biarc with at most two semicircles. More strongly, every st-planar directed graph (a planar directed acyclic graph with a single source and a single sink, both on the outer face) has an arc diagram in which every edge forms a monotonic curve, with these curves all consistently oriented from one end of the vertex line towards the other. For undirected planar graphs, one way to construct an arc diagram with at most two semicircles per edge is to subdivide the graph and add extra edges so that the resulting graph has a Hamiltonian cycle (and so that each edge is subdivided at most once), and to use the ordering of the vertices on the Hamiltonian cycle as the ordering along the line. In a planar graph with $n$ vertices, at most $n/2$ biarcs are needed.

Minimizing crossings

Because it is NP-complete to test whether a given graph has an arc diagram with one semicircle per edge and no crossings, it is also NP-hard to find an arc diagram of this type that minimizes the number of crossings. This crossing minimization problem remains NP-hard, for non-planar graphs, even if the ordering of the vertices along the line is fixed. However, in the fixed-ordering case, an embedding without crossings (if one exists) may be found in polynomial time by translating the problem into a 2-satisfiability problem, in which the variables represent the placement of each arc and the constraints prevent crossing arcs from being placed on the same side of the vertex line. Additionally, in the fixed-ordering case, a crossing-minimizing embedding may be approximated by solving a maximum cut problem in an auxiliary graph that represents the semicircles and their potential crossings (or equivalently, by approximating the MAX2SAT version of the 2-satisfiability instance).

Cimikowski & Shope (1996), Cimikowski (2002), and He, Sýkora & Vrt'o (2005) discuss heuristics for finding arc diagrams with few crossings.

Clockwise orientation

For drawings of directed graphs, a common convention is to draw each arc in a clockwise direction, so that arcs that are directed from an earlier to a later vertex in the sequence are drawn above the vertex line, and arcs directed from a later to an earlier vertex are drawn below the line. This clockwise orientation convention was developed as part of a different graph drawing style by Fekete et al. (2003), and applied to arc diagrams by Pretorius & van Wijk (2007).

Applications

A Farey diagram

The Farey diagram of a set of rational numbers is a structure that may be represented geometrically as an arc diagram. In this form it has a vertex for each number, placed on the number line, and a semicircular edge above the line connecting pairs of numbers $p/q$ and $r/s$ (in simplest terms) for which $|ps-rq|=1$. The semicircles of the diagram may be thought of as lines in the Poincaré half-plane model of the hyperbolic plane, with the vertices placed at infinite points on the boundary line of this model. The Poincaré half-plane model has an infinite point that is not represented as point on the boundary line, the shared endpoint of all vertical rays in the model, and this may be represented by the "fraction" 1/0 (undefined as a number), with the same rule for determining its adjacencies. The Farey diagram of any set of rational numbers is a planar graph, and the Farey diagram of the set of all rational numbers forms a tessellation of the hyperbolic plane by ideal triangles.

Arc diagrams or circuit diagrams are commonly used in studying folded biopolymers such as proteins and nucleic acids (DNAs, RNAs). Biopolymers are typically represented by their primary monomer sequence along the line of the diagrams, and with arcs above the line representing bonds between monomers (e.g., amino acids in proteins or bases in RNA or DNA) that are adjacent in the physical structure of the polymer despite being nonadjacent in the sequence order. The theoretical framework of circuit topology is then typically applied to extract local and global topological information, which can in turn be related to biological function of the folded molecules. When arcs do not cross, the arrangement of the two arcs will be either parallel (P) or series (S). When there are crossings, the crossings represent what is often called as X arrangement in circuit topology. The statistics of P, S, and X can be used to learn about folding kinetics of these polymers.

Arc diagrams were used by Brandes (1999) to visualize the state diagram of a shift register, by Djidjev & Vrt'o (2002) to show that the crossing number of every graph is lower-bounded by a combination of its cutwidth and vertex degrees, by Byrne et al. (2007) to visualize interactions between Bluetooth devices, and by Owens & Jankun-Kelly (2013) to visualize the yardage of plays in a game of American football. Additional applications of this visualization technique are surveyed by Nagel & Duval (2013).

References

• Bekos, Michael A.; Kaufmann, Michael; Kobourov, Stephen G.; Symvonis, Antonios (2013), "Smooth orthogonal layouts", Graph Drawing: 20th International Symposium, GD 2012, Redmond, WA, USA, September 19–21, 2012, Revised Selected Papers, Lecture Notes in Computer Science, vol. 7704, Springer, pp. 150–161, doi:10.1007/978-3-642-36763-2_14, ISBN 978-3-642-36762-5.
• Bernhart, Frank R.; Kainen, Paul C. (1979), "The book thickness of a graph", Journal of Combinatorial Theory, Series B, 27 (3): 320–331, doi:10.1016/0095-8956(79)90021-2.
• Brandes, Ulrik (1999), "Hunting down Graph B", Graph Drawing: 7th International Symposium, GD'99, Štiřín Castle, Czech Republic September 15–19, 1999, Proceedings, Lecture Notes in Computer Science, vol. 1731, Springer, pp. 410–415, doi:10.1007/3-540-46648-7_42, ISBN 978-3-540-66904-3.
• Byrne, Daragh; Lavelle, Barry; Jones, Gareth J. F.; Smeaton, Alan F. (2007), "Visualising Bluetooth interactions: combining the Arc diagram and DocuBurst techniques" (PDF), in Ormerod, Thomas C.; Sas, Corina (eds.), Proceedings of the 21st British HCI Group Annual Conference on HCI 2007: HCI...but not as we know it - Volume 2, BCS HCI 2007, University of Lancaster, United Kingdom, 3-7 September 2007, British Computer Society, pp. 129–132.
• Cardinal, Jean; Hoffmann, Michael; Kusters, Vincent; Tóth, Csaba D.; Wettstein, Manuel (2018), "Arc diagrams, flip distances, and Hamiltonian triangulations", Computational Geometry, 68: 206–225, arXiv:1611.02541, doi:10.1016/j.comgeo.2017.06.001, MR 3715053, S2CID 1169465
• Chung, Fan R. K.; Leighton, Frank Thompson; Rosenberg, Arnold L. (1987), "Embedding graphs in books: A layout problem with applications to VLSI design" (PDF), SIAM Journal on Algebraic and Discrete Methods, 8 (1): 33–58, doi:10.1137/0608002.
• Cimikowski, Robert (2002), "Algorithms for the fixed linear crossing number problem", Discrete Applied Mathematics, 122 (1–3): 93–115, doi:10.1016/S0166-218X(01)00314-6, MR 1907825.
• Cimikowski, Robert; Mumey, Brendan (2007), "Approximating the fixed linear crossing number", Discrete Applied Mathematics, 155 (17): 2202–2210, doi:10.1016/j.dam.2007.05.009, MR 2360650.
• Cimikowski, Robert; Shope, Paul (1996), "A neural-network algorithm for a graph layout problem", IEEE Transactions on Neural Networks, 7 (2): 341–345, doi:10.1109/72.485670, PMID 18255588.
• Djidjev, Hristo; Vrt'o, Imrich (2002), "An improved lower bound for crossing numbers", Graph Drawing: 9th International Symposium, GD 2001, Vienna, Austria, September 23–26, 2001, Revised Papers, Lecture Notes in Computer Science, vol. 2265, Springer, pp. 96–101, doi:10.1007/3-540-45848-4_8, ISBN 978-3-540-43309-5.
• Efrat, Alon; Erten, Cesim; Kobourov, Stephen G. (2007), "Fixed-location circular arc drawing of planar graphs", Journal of Graph Algorithms and Applications, 11 (1): 145–164, doi:10.7155/jgaa.00140.
• Fekete, Jean-Daniel; Wang, David; Dang, Niem; Aris, Aleks; Plaisant, Catherine (2003), "Overlaying graph links on treemaps", IEEE Symp. on Information Visualization, Poster Compendium, pp. 82–83.
• Gilman, Jane; Keen, Linda (2002), "Word sequences and intersection numbers" (PDF), Complex manifolds and hyperbolic geometry (Guanajuato, 2001), Contemporary Mathematics, vol. 311, Providence, Rhode Island: American Mathematical Society, pp. 231–249, doi:10.1090/conm/311/05455, MR 1940172; see Section 2.4, "Farey diagrams and continued fractions"
• Giordano, Francesco; Liotta, Giuseppe; Mchedlidze, Tamara; Symvonis, Antonios (2007), "Computing upward topological book embeddings of upward planar digraphs", Algorithms and Computation: 18th International Symposium, ISAAC 2007, Sendai, Japan, December 17-19, 2007, Proceedings, Lecture Notes in Computer Science, vol. 4835, Springer, pp. 172–183, doi:10.1007/978-3-540-77120-3_17, ISBN 978-3-540-77118-0.
• He, Hongmei; Sýkora, Ondrej; Vrt'o, Imrich (2005), "Crossing Minimisation Heuristics for 2-page Drawings", Electronic Notes in Discrete Mathematics, 22: 527–534, doi:10.1016/j.endm.2005.06.088.
• Heer, Jeffrey; Bostock, Michael; Ogievetsky, Vadim (2010), "A tour through the visualization zoo", Communications of the ACM, 53 (6): 59–67, doi:10.1145/1743546.1743567.
• Kabakçıoğlu, A.; Stella, A. L. (November 2005), "Scale-free network hidden in a collapsing polymer", Physical Review E, 72 (5), 055102(R), arXiv:cond-mat/0409584, Bibcode:2005PhRvE..72e5102K, doi:10.1103/physreve.72.055102, PMID 16383674, S2CID 29977757
• Masuda, Sumio; Nakajima, Kazuo; Kashiwabara, Toshinobu; Fujisawa, Toshio (1990), "Crossing minimization in linear embeddings of graphs", IEEE Transactions on Computers, 39 (1): 124–127, doi:10.1109/12.46286, MR 1032144.
• Nagel, Till; Duval, Erik (2013), "A visual survey of arc diagrams" (PDF), 2013 VIS Posters, IEEE
• Nicholson, T. A. J. (1968), "Permutation procedure for minimising the number of crossings in a network", Proceedings of the Institution of Electrical Engineers, 115: 21–26, doi:10.1049/piee.1968.0004, MR 0311416.
• Owens, Sean Gabriel; Jankun-Kelly, T. J. (2013), "Visualizations for exploration of American football season and play data" (PDF), 1st IEEE VIS Workshop on Sports Data Visualization, IEEE
• Pretorius, A. J.; van Wijk, J. J. (2007), "Bridging the semantic gap: Visualizing transition graphs with user-defined diagrams", IEEE Computer Graphics and Applications, 27 (5): 58–66, doi:10.1109/MCG.2007.121, PMID 17913025, S2CID 8643133.
• Saaty, Thomas L. (1964), "The minimum number of intersections in complete graphs", Proceedings of the National Academy of Sciences of the United States of America, 52 (3): 688–690, Bibcode:1964PNAS...52..688S, doi:10.1073/pnas.52.3.688, MR 0166772, PMC 300329, PMID 16591215.
• Wattenberg, M. (2002), "Arc diagrams: visualizing structure in strings", Proc. IEEE Symposium o nInformation Visualization (INFOVIS 2002), pp. 110–116, doi:10.1109/INFVIS.2002.1173155, ISBN 0-7695-1751-X, S2CID 881989.