Evidence: (1) An indication, for each edge and each vertex of G, whether that edge or vertex should be included in the subgraph; (2) A renumbering of the vertices of H so that H becomes identical to the subgraph indicated by (1).

The evidence is accepted if: The subgraph of G indicated by part (1) of the evidence is identical to the graph obtained by renumbering the vertices of H according to part (2) of the evidence.

Note. The following does not work.

Evidence: An indication, for each edge and each vertex of G, whether that edge or vertex should be included in the subgraph;

The evidence is accepted if: The subgraph of G indicated by the evidence is isomorphic to H.

Remember that the evidence checker must be a polynomial-time algorithm. Nobody knows a polynomial-time algorithm to determine whether two given graphs are isomorphic.

But testing whether two graphs are identical is easy. Check that they have the same vertices and the same edges. There is no renumbering of the vertices.