Let G(V,E) be a connected graph and W{w 1,w 2,…,w k} an ordered set of V. Given v∈V, the representation of v with respect to W is the k-vector r(v|W)(d(v,w 1),d(v,w 2),…,d(v,w k)). The set W is a resolving set of G...Let G(V,E) be a connected graph and W{w 1,w 2,…,w k} an ordered set of V. Given v∈V, the representation of v with respect to W is the k-vector r(v|W)(d(v,w 1),d(v,w 2),…,d(v,w k)). The set W is a resolving set of G if r(u|W)r(v|W) implies that uv for all pairs {u,v} of vertices of G. The resolving set of G with the smallest cardinality is called a basis of G. The dimension of G, dim (G), is the cardinality of a basis for G. The bound of a Cartesian product of a connected graph H and a path P k was reached: dim(H)≤dim(H×P k)≤dim(H)+1. Then, the dimension value of some graphs was given. At last, the constructions of some graphs’ bases were showed.展开更多
文摘Let G(V,E) be a connected graph and W{w 1,w 2,…,w k} an ordered set of V. Given v∈V, the representation of v with respect to W is the k-vector r(v|W)(d(v,w 1),d(v,w 2),…,d(v,w k)). The set W is a resolving set of G if r(u|W)r(v|W) implies that uv for all pairs {u,v} of vertices of G. The resolving set of G with the smallest cardinality is called a basis of G. The dimension of G, dim (G), is the cardinality of a basis for G. The bound of a Cartesian product of a connected graph H and a path P k was reached: dim(H)≤dim(H×P k)≤dim(H)+1. Then, the dimension value of some graphs was given. At last, the constructions of some graphs’ bases were showed.
