# Proof

Let $$G(V,E,\gamma)$$ be a biconnected graph and let $$C(V_c,E_c)$$ be a non-separating cycle with the piece $$P(V_p,E_p)$$, which is, by hypothesis, not a path. Let $$u$$ and $$v$$ be two attachments of $$P$$ that are consecutive in the circular order of $$C$$, i.e. if $$C:=x_0x_1\ldots x_{k-1}x_k$$ with $$x_k=x_0$$, then $$u=x_i$$ and $$v=x_j$$ for some $$i < j$$ and there is no other attachment of $$P$$ among the vertices $$u=x_i,x_{i+1},\ldots,x_{j-1},x_j=v$$, besides $$u$$ and $$v$$.

In the example below, the edges of the cycle $$C$$ are colored blue and the vertices of the piece $$P$$ are colored orange. Moreover, $$u$$ and $$v$$ are two attachments of $$P$$ that are consecutive in the circular order of $$C$$:

We want to show that, provided $$P$$ is not a path (which is the case in our example), it is possible to construct a separating cycle $$S(V_s,E_s)$$, which consists of a subpath of $$C$$ plus a path of $$P$$ between the attachments $$u$$ and $$v$$ of $$P$$.

We observe that, by construction, $$E=x_ix_{i+1}\ldots x_{j-1}x_j$$ is a subpath of $$C$$ from $$u$$ to $$v$$ that does not contain any attachment of $$P$$. Moreover, by definition of a piece, $$P(V_p,E_p)$$ is connected. Thus, there is also another path $$F$$ from $$u$$ to $$v$$ in $$P$$.

In our example, both paths $$E$$ (dashed red) and $$F$$ (dashed blue) are shown in the following figure:

Let $$S$$ be a cycle obtained from $$C$$ by replacing $$E$$ with $$F$$ (see next figure):

It remains to be shown that $$S$$ is a separating cycle cycle. First of all, we have by construction that $$E$$ is a piece of $$G$$ with respect to $$S$$. By hypothesis, $$P$$ is not a path. Therefore, there exists an edge $$e$$ of $$P$$ which is not an edge of $$F$$. Therefore, there must exist another piece of $$G$$ with respect to $$S$$, which is distinct to $$E$$ (see next figure):

Since there are more than one pieces of $$G$$ with respect to $$S$$, $$S$$ is a separating cycle.

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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