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.

A peripheral cycle ${\displaystyle C}$ in a graph ${\displaystyle G}$ can be defined formally in one of several equivalent ways:

The equivalence of these definitions is not hard to see: a connected subgraph of ${\displaystyle G\setminus C}$ (together with the edges linking it to ${\displaystyle C}$), 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 ${\displaystyle C}$.

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