Forcing on perfect matchings - A survey

The notions of a forcing edge and the forcing number of a perfect matching first appeared in a 1991 paper [15] by Harary, Klein and Živković. The root of these concepts can be traced to the works ([24] and [36]) by Randić and Klein in 1985-1987, where the forcing number was introduced under the name of "innate degree of freedom" of a Kekulé structure, which plays an important role in the resonance theory in chemistry. Over the past two decades, more and more mathematicians were attracted to the study on forcing sets (including forcing edges and forcing faces, etc.) and the forcing numbers of perfect matchings of a graph. The scope of graphs in consideration has been extended from polyhexes to various bipartite graphs and non-bipartite graphs. Some varied topics such as global forcing matchings and antiforcing matchings also emerged recently. Here we will present a brief survey on the known results, as well as some open problems and conjectures in this growing field.