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

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.

💡 Research Summary

The paper investigates the deep connections between resonance graphs of plane bipartite graphs, daisy cubes, hypercubes, and independent sets. The authors begin by considering a plane elementary bipartite graph G whose infinite face is forcing—that is, the boundary of the outer face is alternating with respect to some perfect matching. Under this assumption they prove a bijection between the set of maximal hypercubes of the resonance graph R(G) and the set of maximal resonant sets of G. A resonant set is a collection of pairwise vertex‑disjoint finite faces that are simultaneously resonant with respect to a single perfect matching; a maximal resonant set cannot be enlarged while preserving this property. The bijection shows that each maximal hypercube in R(G) corresponds uniquely to a maximal resonant set, extending earlier results that were limited to elementary benzenoid graphs without “nice coronene” structures.

Next the authors focus on a special subclass called peripherally 2‑colorable graphs. These graphs have all vertices of degree 2 or 3, every degree‑3 vertex lies on the outer boundary, and a proper black‑white coloring exists such that degree‑3 vertices alternate in color around the periphery. For such graphs the inner dual G* is a tree whose vertices represent the finite faces of G. Using the previously established hypercube‑resonant‑set correspondence, they demonstrate a bijection between maximal hypercubes of R(G) and maximal independent sets of the tree G*. Consequently, the combinatorial structure of hypercubes inside the resonance graph is completely captured by independent sets of a tree, providing a new bridge between two seemingly unrelated graph concepts.

The paper then turns to daisy cubes, a family of partial cubes defined as induced subgraphs of an n‑dimensional hypercube Qₙ whose vertex set is a downward‑closed subset of the Boolean lattice Bₙ. The authors prove that the resonance graph R(G) of a plane bipartite graph is a daisy cube if and only if it can be expressed as the simplex graph K(Ĝ) of the complement Ĝ of a forest. The simplex graph K(H) has as vertices all cliques of H (including the empty set) and edges join cliques that differ by exactly one vertex; equivalently, K(Ĝ) can be viewed as the graph of independent sets of Ĝ with adjacency defined by a single‑vertex change. Thus, R(G) being a daisy cube is equivalent to the complement of the underlying graph being a forest, and for elementary bipartite graphs this reduces to the complement of a tree. This characterization unifies several known families: Fibonacci cubes Γₙ and Lucas cubes Λₙ appear as K(Ĝ) where Ĝ is the complement of a path Pₙ or a cycle Cₙ, respectively.

Finally, the authors classify all trees that have at most five maximal independent sets. By exhaustive structural analysis they identify precisely those trees whose complement’s simplex graph yields a daisy cube with at most five maximal vertices. Consequently, every daisy cube with ≤ 5 maximal vertices that is also a simplex graph of a complement of a tree is isomorphic to the resonance graph of some plane elementary bipartite graph. The paper also revisits known properties of Fibonacci cubes, providing alternative proofs via the new framework.

In summary, the main contributions are:

1. A bijection between maximal hypercubes of R(G) and maximal resonant sets of G for any plane elementary bipartite graph with a forcing outer face.
2. For peripherally 2‑colorable graphs, a bijection between maximal hypercubes of R(G) and maximal independent sets of the inner‑dual tree.
3. A necessary and sufficient condition for a resonance graph to be a daisy cube: it must be the simplex graph of the complement of a forest (or of a tree in the elementary case).
4. A complete classification of trees with ≤ 5 maximal independent sets, leading to a full description of daisy cubes with ≤ 5 maximal vertices that arise as resonance graphs.

These results deepen the understanding of how resonance structures in chemical graph theory relate to well‑studied combinatorial objects, and they open new avenues for applying hypercube and independent‑set techniques to the analysis of molecular resonance patterns.