Before revision: In this paper, we introduce a new problem called Tree-Residue Vertex-Breaking (TRVB): given a multigraph G some of whose vertices are marked "breakable," is it possible to convert G into a tree via a sequence of "vertex-breaking" operations ( disconnecting the edges at a degree-k breakable vertex by replacing that vertex with k degree-1 vertices  )?
After revision: In this paper, we introduce a new problem called Tree-Residue Vertex-Breaking (TRVB): given a multigraph G some of whose vertices are marked "breakable," is it possible to convert G into a tree via a sequence of "vertex-breaking" operations ( replacing a degree-k breakable vertex by replacing that vertex with k degree-1 vertices )?