= Solution
\b[No, it cannot deformation retract onto a <graph>.] Each <sphere> contributes a degree-$2$ <homology (mathematics)> class independent of the other two. Attaching <circles> at points introduces only vertices and edges, so
$$
\boxed{H_2(\widehat X;\mathbb Z)\cong\mathbb Z^3.}
$$
For example, the <cellular chain complex> obtained by giving each <sphere> one $2$-cell and a <tree> of arcs has three zero-boundary $2$-cells and no $3$-cells. Every <graph> has zero degree-$2$ <homology (mathematics)>. A <deformation retraction> would induce a <homotopy equivalence> and an isomorphism on <homology (mathematics)>, which is impossible. An ordinary <retraction> has no such requirement.
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