On forcing matching number of boron-nitrogen fullerene graphs

Let G be a graph that admits a perfect matching M. A forcing set S for a perfect matching M is a subset of M such that it is contained in no other perfect matchings of G. The smallest cardinality of forcing sets of M is called the forcing number of M. Computing the minimum forcing number of perfect matchings of a graph is an NP-complete problem. In this paper, we consider boron-nitrogen (BN) fullerene graphs, cubic 3-connected plane bipartite graphs with exactly six square faces and other hexagonal faces. We obtain the forcing spectrum of tubular BN-fullerene graphs with cyclic edge-connectivity 3. Then we show that all perfect matchings of any BN-fullerene graphs have the forcing number at least two. Furthermore, we mainly construct all seven BN-fullerene graphs with the minimum forcing number two.

[1]  Heping Zhang,et al.  Forcing matching numbers of fullerene graphs , 2010, Discret. Appl. Math..

[2]  D. Vukicevic,et al.  Total forcing number of the triangular grid , 2004 .

[3]  Pierre Hansen,et al.  Bonds Fixed by Fixing Bonds , 1993, J. Chem. Inf. Comput. Sci..

[4]  Heping Zhang,et al.  Plane elementary bipartite graphs , 2000, Discret. Appl. Math..

[5]  Heping Zhang,et al.  The forcing number of toroidal polyhexes , 2008 .

[6]  Heping Zhang,et al.  A NEW ENUMERATION METHOD FOR KEKULE STRUCTURES OF HEXAGONAL SYSTEMS WITH FORCING EDGES , 1995 .

[7]  Tomislav Došlić,et al.  Cyclical Edge-Connectivity of Fullerene Graphs and (k, 6)-Cages , 2003 .

[8]  Lior Pachter,et al.  Forcing matchings on square grids , 1998, Discret. Math..

[9]  Lior Pachter,et al.  Forcing numbers of stop signs , 2003, Theor. Comput. Sci..

[10]  Xueliang Li.  Hexagonal Systems with Forcing Single Edges , 1997, Discret. Appl. Math..

[11]  T. Došlić.  Global Forcing Number of Benzenoid Graphs , 2007 .

[12]  M D Plummer.  Matching Extension and Connectivity in Graphs. 1. Introduction and Terminology , 1986 .

[13]  F. Harary,et al.  Graphical properties of polyhexes: Perfect matching vector and forcing , 1991 .

[14]  Hamed Hatami,et al.  On the spectrum of the forced matching number of graphs , 2004, Australas. J Comb..

[15]  Tomislav Doslic,et al.  Global forcing number of grid graphs , 2007, Australas. J Comb..

[16]  Seth Kleinerman.  Bounds on the forcing numbers of bipartite graphs , 2006, Discret. Math..

[17]  Xueliang Li,et al.  Hexagonal systems with forcing edges , 1995, Discret. Math..

[18]  Peter Adams,et al.  On the forced matching numbers of bipartite graphs , 2004, Discret. Math..

[19]  Matthew E. Riddle.  The minimum forcing number for the torus and hypercube , 2002, Discret. Math..

[20]  李学良,et al.  FORCING BONDS OF A BENZENOID SYSTEM , 1996 .

[21]  Li Xueliang,et al.  Forcing bonds of a benzenoid system , 1996 .

[22]  M. Randic,et al.  Innate degree of freedom of a graph , 1987 .