Homology classification of spatial graphs by linking numbers and Simon invariants

We show that two embeddings f and g of a finite graph G into the 3-space are spatial-graph-homologous if and only if for each subgraph H of G that is homeomorphic to a disjoint union of two circles, the restriction maps f|H and g|H have the same linking number, and for each subgraph H of G that is homeomorphic to a complete graph K5 or a complete bipartite graph K3,3, the restriction maps f|H and g|H have the same Simon invariant.