Clar structures vs Fries structures in hexagonal systems

A hexagonal system H is a 2-connected bipartite plane graph such that all inner faces are hexagons, which is often used to model the structure of a benzenoid hydrocarbon or graphen. A perfect matching of H is a set of disjoint edges which covers all vertices of H. A resonant set S of H is a set of hexagons in which every hexagon is M-alternating for some perfect matching M. The Fries number of H is the size of a maximum resonant set and the Clar number of H is the size of a maximum independent resonant set (i.e. all hexagons are disjoint). A pair of hexagonal systems with the same number of vertices is called a contra-pair if one has a larger Clar number but the other has a larger Fries number. In this paper, we investigates the Fries number and Clar number for hexagonal systems, and show that a catacondensed hexagonal system has a maximum resonant set containing a maximum independent resonant set, which is conjectured for all hexagonal systems. Further, our computation results demonstrate that there exist many contra-pairs.