Resonance graphs that are daisy cubes: from hypercubes to independent sets via resonant sets

TL;DR

This paper establishes a bijection between maximal hypercubes in resonance graphs of certain plane bipartite graphs and maximal resonant sets, linking graph theory concepts with chemical resonance structures.

Contribution

It generalizes previous results by connecting resonance graphs with hypercubes and independent sets, and characterizes when resonance graphs are daisy cubes.

Findings

Bijection between maximal hypercubes and resonant sets for plane elementary bipartite graphs.

Resonance graph is a daisy cube iff it is the simplex graph of a forest's complement.

Characterization of trees with up to five maximal independent sets for daisy cube identification.

Abstract

Let $G$ be a plane elementary bipartite graph whose infinite face is forcing. We provide a bijection between the set of maximal hypercubes of its resonance graph and the set of maximal resonant sets of $G$, which generalizes a main result in [MATCH Commun. Math. Comput. Chem. 68 (2012) 65-77], where $G$ was only considered as an elementary benzenoid graph without nice coronenes. For a special case when $G$ is a peripherally 2-colorable graph, it follows that there is a bijection between the set of maximal hypercubes of its resonance graph and the set of maximal independent sets of a tree that is the inner dual of $G$. We then show that the resonance graph of a plane bipartite graph $G$ is a daisy cube if and only if it is the simplex graph of the complement of a forest. Finally, we characterize trees with at most 5 maximal independent sets to determine daisy cubes that are simplex graphs of the complements of trees and having at most five maximal vertices.