An even cycle C in a graph G is called a conjugated circuit if a perfect matching of C can be extended to a perfect matching of G. An edge of a graph G is called a forcing edge if it is contained in exactly one perfect matching of G. The two concepts have played important roles in the studies of Kekulé structures of benzenoid hydrocarbon molecules. In this paper, we first present a different proof for a result just published in the 2012 article  by Klavžar and Salem, that is, all circuits of a 2-connected outerplanar bipartite graph are conjugated. Then we further give a characterization for the conjugated circuits in any 2-connected (no matter whether bipartite or not) outerplanar graph with an even number of vertices. We also show that each edge of a connected bipartite graph G is a forcing edge if and only if G is an even cycle or an edge, which generalizes a main result in the 1991 paper  on polyhexes by Harary, Klein, and Živković. Finally we present miscellaneous related results on perfect matching forcings, one of which asserts that a bipartite graph G with 2n vertices has its forcing number attaining the largest possible value n-1 if and only if G is the complete bipartite graph Kn,n.
|Original language||English (US)|
|Number of pages||12|
|State||Published - 2013|
All Science Journal Classification (ASJC) codes
- Computer Science Applications
- Computational Theory and Mathematics
- Applied Mathematics