Enumeration of spatial 2-bouquet graphs up to flat vertex isotopy

We enumerate spatial 2-bouquet graphs, or spatial graphs having exactly one 4-valent vertex and no other vertices, up to flat vertex isotopy. In order to do that, we give a method of constructing all such graphs from 2-string tangles, and distinguish the resulting graphs by computing their Yamada polynomials. We then prove that there exist exactly 51 flat vertex isotopy classes of the prime spatial 2-bouquet graphs with up to six crossings.