Isomorphic daisy cubes based on their τ-graphs

We prove that if A and B are daisy cubes whose τ -graphs are forests, then A and B are isomorphic if and only if their τ -graphs are isomorphic. The result is applied to show that a daisy cube with at least one edge is the resonance graph of a plane bipartite graph G if and only if its τ -graph is a forest which is isomorphic to the inner dual of the subgraph of G obtained by removing all forbidden edges. As a consequence, some well-known properties of the τ -graphs of Fibonacci cubes and Lucas cubes are provided as examples with different proofs. Open questions for the characterizations of Lucas cubes in terms of their τ -graphs are raised.