Induced-Cover Excision Characterizes Dismantlable Hom Tests

Fix a nonempty finite graph T, with loops permitted, and a nonzero commutative coefficient ring R with identity. We classify when ordinary relative homology of Hom(T,-) satisfies excision for every induced graph cover. This occurs exactly when T dismantles to a single looped vertex. Exactness of the degree-zero Mayer-Vietoris sequence for all such covers already characterizes these tests. Otherwise, an explicit obstruction uses at most |V(T)|+1 target vertices and its defect contains a split copy of R. The proof combines the established fold theorem and isolated identity of a stiff graph with an unlooped-twin or nonedge-deletion construction. For the edge test K2, the two-edge cover of a three-vertex loopless path is a counterexample. Exact finite checks cover 1,098 tests, 502 obstructions and 46,758 induced clique covers; a separate cellular-boundary implementation replays 31 small negative cases. This is a complete classification of the specified fixed-test family, not a full resolution of the broader AIM graph-homology program. Classical inputs are credited. The preprint is self-audited and unrefereed, with no independent-review, formal-verification or absolute-priority claim.