关于一些特殊图类的弱控制多项式的研究

Research on Weak Domination Polynomials of Some Special Graph Classes

研究一些特殊图类的弱控制多项式.令图G=(V(G),E(G))是一个简单连通图,若对任意v∈V(G),存在u∈V(G),使得uv∈E(G)且d(u)≥d(v)成立,则称v弱控制u.设W(G)?V(G),如果对任意u∈V(G)W(G),存在v∈W(G),使得v弱控制u,则称W(G)为图G的一个弱控制集.含点数最少的弱控制集称为最小弱控制集,最小弱控制集中所包含点的个数称为图G的弱控制数,记为γwd(G).图G的弱控制多项式为WD(G,x)=nΣj=γwd(G)Wd(G,j)x印j,其中Wd(G,j)表示图G中阶为j的弱控制集的个数.

Let G=(V(G),E(G))be a connected simple graph.If for any vertex v∈V(G),there exist a vertex u∈V(G)such that uv∈E(G)and d(u)≥d(v),then v weakly dominate u.Let W(G)V(G),if for every vertex u∈V(G)\W(G),there exists at least one vertex v∈W(G)so that v weakly dominate u,then W(G)is called a weak dominating set of graph G.The minimum weak dominating set is a weak dominating set with the least number of vertices.The minimum cardinality of such a set is called the weak domination number of G and it is denoted byγwd(G).The weak domination polynomial of graph G is calculated using WD(G,x)=nΣj=γwd(G)Wd(G,j)xj,where Wd(G,j)is the number of weak dominating set of G of order j.In this paper we study the weak domination polynomials of some special graph classes.