# Peripheral cycle

In graph theory, a **peripheral cycle** (or **peripheral circuit**) in an undirected graph is, intuitively, a cycle that does not separate any part of the graph from any other part. Peripheral cycles (or, as they were initially called, **peripheral polygons**, because Tutte called cycles "polygons") were first studied by Tutte (1963), and play important roles in the characterization of planar graphs and in generating the cycle spaces of nonplanar graphs.[1]

## Definitions

A peripheral cycle in a graph can be defined formally in one of several equivalent ways:

- is peripheral if it is a simple cycle in a connected graph with the property that, for every two edges and in , there exists a path in that starts with , ends with , and has no interior vertices belonging to .[2]
- is peripheral if it is an induced cycle with the property that the subgraph formed by deleting the edges and vertices of is connected.[3]
- If is any subgraph of , a
*bridge*[4] of is a minimal subgraph of that is edge-disjoint from and that has the property that all of its points of attachments (vertices adjacent to edges in both and ) belong to .[5] A simple cycle is peripheral if it has exactly one bridge.[6]

The equivalence of these definitions is not hard to see: a connected subgraph of (together with the edges linking it to ), or a chord of a cycle that causes it to fail to be induced, must in either case be a bridge, and must also be an equivalence class of the binary relation on edges in which two edges are related if they are the ends of a path with no interior vertices in .[7]

## Properties

Peripheral cycles appear in the theory of polyhedral graphs, that is, 3-vertex-connected planar graphs. For every planar graph , and every planar embedding of , the faces of the embedding that are induced cycles must be peripheral cycles. In a polyhedral graph, all faces are peripheral cycles, and every peripheral cycle is a face.[8] It follows from this fact that (up to combinatorial equivalence, the choice of the outer face, and the orientation of the plane) every polyhedral graph has a unique planar embedding.[9]

In planar graphs, the cycle space is generated by the faces, but in non-planar graphs peripheral cycles play a similar role: for every 3-vertex-connected finite graph, the cycle space is generated by the peripheral cycles.[10] The result can also be extended to locally-finite but infinite graphs.[11] In particular, it follows that 3-connected graphs are guaranteed to contain peripheral cycles. There exist 2-connected graphs that do not contain peripheral cycles (an example is the complete bipartite graph , for which every cycle has two bridges) but if a 2-connected graph has minimum degree three then it contains at least one peripheral cycle.[12]

Peripheral cycles in 3-connected graphs can be computed in linear time and have been used for designing planarity tests.[13] They were also extended to the more general notion of non-separating ear decompositions. In some algorithms for testing planarity of graphs, it is useful to find a cycle that is not peripheral, in order to partition the problem into smaller subproblems. In a biconnected graph of circuit rank less than three (such as a cycle graph or theta graph) every cycle is peripheral, but every biconnected graph with circuit rank three or more has a non-peripheral cycle, which may be found in linear time.[14]

Generalizing chordal graphs, Seymour & Weaver (1984) define a strangulated graph to be a graph in which every peripheral cycle is a triangle. They characterize these graphs as being the clique-sums of chordal graphs and maximal planar graphs.[15]

## Related concepts

Peripheral cycles have also been called non-separating cycles,[2] but this term is ambiguous, as it has also been used for two related but distinct concepts: simple cycles the removal of which would disconnect the remaining graph,[16] and cycles of a topologically embedded graph such that cutting along the cycle would not disconnect the surface on which the graph is embedded.[17]

In matroids, a non-separating circuit is a circuit of the matroid (that is, a minimal dependent set) such that deleting the circuit leaves a smaller matroid that is connected (that is, that cannot be written as a direct sum of matroids).[18] These are analogous to peripheral cycles, but not the same even in graphic matroids (the matroids whose circuits are the simple cycles of a graph). For example, in the complete bipartite graph , every cycle is peripheral (it has only one bridge, a two-edge path) but the graphic matroid formed by this bridge is not connected, so no circuit of the graphic matroid of is non-separating.

## References

- Tutte, W. T. (1963), "How to draw a graph",
*Proceedings of the London Mathematical Society*, Third Series,**13**: 743–767, doi:10.1112/plms/s3-13.1.743, MR 0158387. - Di Battista, Giuseppe; Eades, Peter; Tamassia, Roberto; Tollis, Ioannis G. (1998),
*Graph Drawing: Algorithms for the Visualization of Graphs*, Prentice Hall, pp. 74–75, ISBN 978-0-13-301615-4. - This is, essentially, the definition used by Bruhn (2004). However, Bruhn distinguishes the case that has isolated vertices from disconnections caused by the removal of .
- Not to be confused with bridge (graph theory), a different concept.
- Tutte, W. T. (1960), "Convex representations of graphs",
*Proceedings of the London Mathematical Society*, Third Series,**10**: 304–320, doi:10.1112/plms/s3-10.1.304, MR 0114774. - This is the definition of peripheral cycles originally used by Tutte (1963). Seymour & Weaver (1984) use the same definition of a peripheral cycle, but with a different definition of a bridge that more closely resembles the induced-cycle definition for peripheral cycles.
- See e.g. Theorem 2.4 of Tutte (1960), showing that the vertex sets of bridges are path-connected, see Seymour & Weaver (1984) for a definition of bridges using chords and connected components, and also see Di Battista et al. (1998) for a definition of bridges using equivalence classes of the binary relation on edges.
- Tutte (1963), Theorems 2.7 and 2.8.
- See the remarks following Theorem 2.8 in Tutte (1963). As Tutte observes, this was already known to Whitney, Hassler (1932), "Non-separable and planar graphs",
*Transactions of the American Mathematical Society*,**34**(2): 339–362, doi:10.2307/1989545, MR 1501641, PMC 1076008. - Tutte (1963), Theorem 2.5.
- Bruhn, Henning (2004), "The cycle space of a 3-connected locally finite graph is generated by its finite and infinite peripheral circuits",
*Journal of Combinatorial Theory*, Series B,**92**(2): 235–256, doi:10.1016/j.jctb.2004.03.005, MR 2099143. - Thomassen, Carsten; Toft, Bjarne (1981), "Non-separating induced cycles in graphs",
*Journal of Combinatorial Theory*, Series B,**31**(2): 199–224, doi:10.1016/S0095-8956(81)80025-1, MR 0630983. - Schmidt, Jens M. (2014),
*The Mondshein Sequence*, pp. 967–978, doi:10.1007/978-3-662-43948-7_80. - Di Battista et al. (1998), Lemma 3.4, pp. 75–76.
- Seymour, P. D.; Weaver, R. W. (1984), "A generalization of chordal graphs",
*Journal of Graph Theory*,**8**(2): 241–251, doi:10.1002/jgt.3190080206, MR 0742878. - E.g. see Borse, Y. M.; Waphare, B. N. (2008), "Vertex disjoint non-separating cycles in graphs",
*The Journal of the Indian Mathematical Society*, New Series,**75**(1–4): 75–92 (2009), MR 2662989. - E.g. see Cabello, Sergio; Mohar, Bojan (2007), "Finding shortest non-separating and non-contractible cycles for topologically embedded graphs",
*Discrete and Computational Geometry*,**37**(2): 213–235, doi:10.1007/s00454-006-1292-5, MR 2295054. - Maia, Bráulio, Junior; Lemos, Manoel; Melo, Tereza R. B. (2007), "Non-separating circuits and cocircuits in matroids",
*Combinatorics, complexity, and chance*, Oxford Lecture Ser. Math. Appl.,**34**, Oxford: Oxford Univ. Press, pp. 162–171, doi:10.1093/acprof:oso/9780198571278.003.0010, MR 2314567.