Generalized Baumslag–Solitar groups are defined as fundamental groups of graphs of groups with infinite cyclic vertex and edge groups. Forester [Geom. Topol. 6 (2002) 219-267] proved that in most cases the defining graphs are cyclic JSJ decompositions, in the sense of Rips and Sela. Here we extend Forester’s results to graphs of groups with vertex groups that can be either infinite cyclic or quadratically hanging surface groups.
"JSJ decompositions of quadratic Baumslag–Solitar groups." Algebr. Geom. Topol. 12 (4) 2027 - 2047, 2012. https://doi.org/10.2140/agt.2012.12.2027