Geodesic
Graph manifolds genus 2
Vestnik Chelyabinskogo Gosudarstvennogo Universiteta. Matematika, Mekhanika, Informatika, no. 10 (2008), pp. 94-100

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We prove one necessary condition for totally orientable graph manifolds genus 2. More precisely, we describe such the set of graphs, that any totally orientable graph manifold genus two is constructed on this graph. 
@article{VCHGU_2008_10_a12,
     author = {F. G. Korablev},
     title = {Graph manifolds genus 2},
     journal = {Vestnik Chelyabinskogo Gosudarstvennogo Universiteta. Matematika, Mekhanika, Informatika},
     pages = {94--100},
     publisher = {mathdoc},
     number = {10},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VCHGU_2008_10_a12/}
}
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F. G. Korablev. Graph manifolds genus 2. Vestnik Chelyabinskogo Gosudarstvennogo Universiteta. Matematika, Mekhanika, Informatika, no. 10 (2008), pp. 94-100. http://geodesic.mathdoc.fr/item/VCHGU_2008_10_a12/