Niko Tratnik: Resonance graphs of plane bipartite graphs as daisy cubes

We characterize plane bipartite graphs whose resonance graphs are daisy cubes, and therefore generalize related results on resonance graphs of benzenoid graphs, catacondensed even ring systems, as well as 2-connected outerplane bipartite graphs. Firstly, we show that if G is a plane elementary bipartite graph other than K₂, then the resonance graph of G is a daisy cube if and only if the Fries number of G equals the number of finite faces of G. Next, we extend the above characterization from plane elementary bipartite graphs to plane bipartite graphs and show that the resonance graph of a plane bipartite graph G is a daisy cube if and only if G is weakly elementary bipartite such that each of its elementary component G_i other than K₂ holds the property that the Fries number of G_i equals the number of finite faces of G_i. Along the way, we provide a structural characterization for a plane elementary bipartite graph whose resonance graph is a daisy cube, and show that a Cartesian product graph is a daisy cube if and only if all of its nontrivial factors are daisy cubes.

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