A decomposition structure of resonance graphs that are daisy cubes

Abstract

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 $R\left(G\right)$ be its resonance graph. We provide a decomposition structure of $R\left(G\right)$ with respect to an arbitrary finite face of $G$ together with a proper labeling for the vertex set of $R\left(G\right)$. An algorithm is obtained to generate a binary coding for all perfect matchings of $G$ which induces an isometric embedding of $R\left(G\right)$ as a daisy cube into an $n$-dimensional hypercube, where $n$ is the isometric dimension of $R\left(G\right)$. We conclude the paper with an easy conversion between two binary codings for all perfect matchings of $G$ which induce distinct structures on $R\left(G\right)$: one as a daisy cube and the other as a finite distributive lattice, respectively.