Geodesic
Generalized graph manifolds and their effective recognition
Sbornik. Mathematics, Tome 189 (1998) no. 10, pp. 1517-1531

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A generalized graph manifold is a three-dimensional manifold obtained by gluing together elementary blocks, each of which is either a Seifert manifold or contains no essential tori or annuli. By a well-known result on torus decomposition each compact three-dimensional manifold with boundary that is either empty or consists of tori has a canonical representation as a generalized graph manifold. A short simple proof of the existence of a canonical representation is presented and a (partial) algorithm for its construction is described. A simple hyperbolicity test for blocks that are not Seifert manifolds is also presented. 
@article{SM_1998_189_10_a4,
     author = {S. V. Matveev},
     title = {Generalized graph manifolds and their effective recognition},
     journal = {Sbornik. Mathematics},
     pages = {1517--1531},
     publisher = {mathdoc},
     volume = {189},
     number = {10},
     year = {1998},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1998_189_10_a4/}
}
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S. V. Matveev. Generalized graph manifolds and their effective recognition. Sbornik. Mathematics, Tome 189 (1998) no. 10, pp. 1517-1531. http://geodesic.mathdoc.fr/item/SM_1998_189_10_a4/