Peripheral Convex Expansions of Resonance Graphs

Abstract

A perfect matching of a graph is a set of independent edges that covers all vertices of the graph. A bipartite graph is elementary if and only if it is connected and each edge is contained in a perfect matching. The resonance graph of a plane bipartite graph G is a graph whose vertices are perfect matchings of G, and two perfect matchings are adjacent if edges contained in their union but not intersection form a cycle surrounding a finite face of G. It is well known that the resonance graph of a plane elementary bipartite graph is a median graph. Median graphs form an important subclass of partial cubes and have a wide range of applications. The most important structural characterization of a median graph is the Mulder’s convex expansion theorem. Fibonacci cubes used in network designs are median graphs and can be obtained from an edge by a sequence of peripheral convex expansions. Plane bipartite graphs whose resonance graphs are Fibonacci cubes were studied by Klavžar et al. first for their applications in chemistry and completely characterized by Zhang et al. later. Motivated from their work, we characterize all plane bipartite graphs whose resonance graphs can be constructed from an edge by a sequence of peripheral convex expansions.