I have seen in several sources that this results holds, however none of them included the proof. Does anyone know where I can find one?


Also, it would be great if someone could provide me with a counterexample, where irreducibility of the manifold matters.


Thank you.



"Well-known fact" that every irreducible 3-manifold with non-empty boundary has an incompressible surface
The proof might be too long for this fact. However, here is one reference


*Algorithmic Topology and Classification of 3-Manifolds by Sergei Matveev in the series Algorithms and computations in Mathematics, Volume 9, 2003, Springer-Verlag.*


You may start reading from page 167.


