A decomposition structure of resonance graphs that are daisy cubes

It has recently been shown in Brezovnik et al. (2025) that the resonance graph of a plane elementary bipartite graph G with more than two vertices is a daisy cube if and only if G is peripherally 2-colorable. Let G be a peripherally 2-colorable graph and R(G) be its resonance graph. We provide a decomposition structure of R(G) with respect to an arbitrary finite face of G together with a proper labeling for the vertex set of R(G). An algorithm is obtained to generate a binary coding for all perfect matchings of G which induces an isometric embedding of R(G) as a daisy cube into an n-dimensional hypercube, where n is the isometric dimension of R(G). We conclude the paper with an easy conversion between two binary codings for all perfect matchings of G which induce distinct structures on R(G): one as a daisy cube and the other as a finite distributive lattice, respectively.