The Forcing Spectra Of Convex Hexagonal Systems And Related Graphs

Let M be a perfect matching of a graph G.A forcing set S for a perfect matching M of G is a subset of M such that it is contained in no other perfect matchings of G.The cardinality of a forcing set of M with the smallest size is called the forcing number of M,denoted by f(G,M).The forcing spectrum of G is defined as:Spec(G)={f(G,M)|M is a perfect matching of G}.The minimum and maximum values in the forcing spectrum of G are denoted as f(G)and F(G),respectivelyA hexagonal system H is a plane 2-connected graph,in which each inner face is a regular hexagon with side length 1.The Z-transformation graph Z(H)is the graph whose vertices are all perfect matchings of H and the two vertices are adjacent if and only if the symmetry difference of their corresponding perfect matchings is a hexagon.Let Hz be a hexagonal system with Z(H)containing a 1-degree vertex and sz be a hexagon determined by the perfect matching corresponding to the vertex of 1-degree in Z(Hz).If Hz includes maximum m)and T(k)with sz as the center,we will denote Hz as Hz(m,k)where m?k?2m.Recently,Zhang and Zhang proved that for a regular convex hexagonal system O(m,m,m),its forcing spectrum is continuous,and obtained the minimum forcing number of hexagonal system Hz(m,k).The main work of this paper is to study the forced spectrum of Hz.In this paper,firstly we prove that the forcing spectrum of convex hexagonal system O(m,k,n)is continuous.Secondly,we obtain that the forcing spectrum of T(nz)is discontinuous and in particular,F(T(m))-1?Spec(T(m));when m is odd,it is proved that its forcing spectrum is only missing value F(T(m))-1;when m is even,it is proved that[f(T(m)),F(T(m))]\{f(T(m))+1,F(T(m))-1}?Spec(T(m)),but a perfect matching of T(m)has not been constructed whose forcing number equal to f(T(m))+1.Finally,for general hexagonal system Hz(m,k),we prove that its forcing spectrum may be missing two values f(Hz(m,k)+ 1 and 1/2k2 +1/2k-1 in the range of[f(Hz(m,k)),1/2k2+1/2k].and give their relevant conjectures.