Peripheral cycle explained

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, and play important roles in the characterization of planar graphs and in generating the cycle spaces of nonplanar graphs.^[1]

Definitions

A peripheral cycle

C in a graph G can be defined formally in one of several equivalent ways:C is peripheral if it is a simple cycle in a connected graph with the property that, for every two edges e _{1} and e _{2} in G\setminus C , there exists a path in G that starts with e _{1} , ends with e _{2} , and has no interior vertices belonging to C .^[2]

• If

C is any subgraph of G , a bridge^[3] of C is a minimal subgraph B of G that is edge-disjoint from C and that has the property that all of its points of attachments (vertices adjacent to edges in both B and G\setminus B ) belong to C .^[4] A simple cycle C is peripheral if it has exactly one bridge.^[5]

• In a connected graph that is not a theta graph, peripheral cycles cannot have chords, because any chord would be a bridge, separated from the rest of the graph. In this case,

C is peripheral if it is an induced cycle with the property that the subgraph G\setminus C formed by deleting the edges and vertices of C is connected.^[6]

The equivalence of these definitions is not hard to see: a connected subgraph of

G\setminus C (together with the edges linking it to C ), 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 C .^[7]

Properties

Peripheral cycles appear in the theory of polyhedral graphs, that is, 3-vertex-connected planar graphs. For every planar and every planar embedding of

G , 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] K _{2,4} , 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, 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]

K _{2,3} , 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 K _{2,3} is non-separating.

Notes and References

1. .
2. .
3. Not to be confused with bridge (graph theory), a different concept.
4. .
5. This is the definition of peripheral cycles originally used by . 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.
6. This is, essentially, the definition used by . However, Bruhn distinguishes the case that G has isolated vertices from disconnections caused by the removal of C .
7. See e.g. Theorem 2.4 of, showing that the vertex sets of bridges are path-connected, see for a definition of bridges using chords and connected components, and also see for a definition of bridges using equivalence classes of the binary relation on edges.
8. , Theorems 2.7 and 2.8.
9. See the remarks following Theorem 2.8 in . As Tutte observes, this was already known to .
10. , Theorem 2.5.
11. .
12. .
13. .
14. , Lemma 3.4, pp. 75–76.
15. .
16. E.g. see .
17. E.g. see .
18. .