# Definition: Pieces of a Graph With Respect to A Cycle

Let $$G(V,E,\gamma)$$ be an biconnected graph and let $$C(V_c,E_c)$$ be a cycle in $$G$$1. The cycle $$C$$ partitions all edges $$E\setminus E_c$$, i.e. the edges, which are not in the cycle, into equivalence classes in the following way:

Two edges $$e_1,e_2\in E\setminus E_c$$ are equivalent if and only if there is a path $$P(V_p,E_p)$$ between them that does not contain any vertex of $$C$$, formally $\forall e_1,e_2\in E\setminus E_c:~e_1\equiv e_2\Longleftrightarrow\exists \text{ path }P:~e_1,e_2\in E_p\wedge V_p\cap V_c=\emptyset.$

The subgraphs of $$G$$ induced by these equivalence classes are called the pieces of the graph $$G$$ with respect to the cycle $$C$$.

The vertices of a piece $$P_i(V_i,E_i)$$ with respect to the cycle $$C(V_c,E_c)$$ that are also in $$C$$, i.e. $$v\in V_i\cap V_c$$, are called the attachments of that piece.

### Example

A graph (left-upper corner) with a cycle (blue) and the corresponding partition of this graph into $$6$$ pieces $$P_1,P_2,\ldots,P_6$$. While $$P_1,P_2,P_5,P_6$$ have two attachments, the pieces $$P_3$$ and $$P_4$$ have three attachments.

Definitions: 1 2
Lemmas: 3
Proofs: 4
Theorems: 5

Thank you to the contributors under CC BY-SA 4.0!

Github:

### 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

1. Please note that in a biconnected graph such a cycle always exists.