Guiradel generalized the notion of geometric intersection number between curves on on a surface to any pair of G–trees; associated to a pair of G–trees, he constructed a closed, 2–dimensional finite CAT(0) cube complex with boundaries whose volume represents the intersection number between the two trees. If the trees are duals to sphere systems one can give a description of the core using the intersection patterns of spheres. We investigate how the core changes along a surgery sequence. We then use this to show that forward and backward surgery paths have a Hausdorff distance of at most 4 in the sphere graph. Speaker(s): Yulan Qing (Toronto)