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.
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.[1]
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.[2] More recently, arc diagrams have been used within the framework of circuit topology of knots and tangles, where they are termed as circuit diagrams.[3]
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.
^Nagel & Duval (2013).
^Masuda et al. (1990).
^Alireza Mashaghi and Roland van der Veen, Polynomial Invariant of Molecular Circuit Topology Symmetry 13(9), 1751 (2021)
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