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Definition: Separating and Non-Separating Cycles

Let $G(V,E,\gamma)$ be an biconnected graph and let $C(V_c,E_c)$ be a cycle in $G$¹.

$C$ is called separating if there are at least two pieces of $G$ with respect to $C$ . $C$ is called non-separating if there only one piece of $G$ with respect to $C$ .

Example

In the following graph, the cycle (blue) is non-separating, since it has only one piece $P$ (highlighted orange on the right).

Another cycle in the same graph (also marked blue) is separating, since it has $6$ pieces $P_1,P_2,\ldots,P_6$ (highlighted orange on the right):

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Lemmas: 1
Proofs: 2

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References

Bibliography

1. Di Battista G., Eades P., Tamassia R., Tollis, I.G.: "Graph Drawing - Algorithms for the Visualization of Graphs", Prentice-Hall, Inc., 1999

Footnotes