Isomorphic daisy cubes based on their $\ddot{I}„$-graphs

Abstract

We prove that if $A$ and $B$ are daisy cubes whose $\ddot{I}„$-graphs are forests, then $A$ and $B$ are isomorphic if and only if their $\ddot{I}„$-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 $\ddot{I}„$-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 Fibonacci cubes and Lucas cubes are provided as examples with different proofs.