Spines and surgery descriptions of graph manifolds

Graph manifolds are compact orientable 3--manifolds obtained by gluing several copies of $D^2 \times \mathbb S^1$ and $N^2 \times \mathbb S^1$ together by homeomorphisms of some components of their boundaries ($D^2$ is the 2--disc and $N^2$ denotes the 2--disc with two holes). Here we study spines and surgery representations of orientable graph manifolds, and derive geometric presentations of their fundamental group. Then we determine the homeomorphism type of many Takahashi manifolds and the Teragaito manifolds, showing that they are graph manifolds with specified invariants. Finally, we describe graph manifolds arising from toroidal surgeries on certain classes of hyperbolic knots.