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
The definitions of generalized pseudoconvex,generalized quasiconvex and its stri ctly generalized convexity were presented for the static programming at locally star -shaped set using the concept of right-upper deriva...The definitions of generalized pseudoconvex,generalized quasiconvex and its stri ctly generalized convexity were presented for the static programming at locally star -shaped set using the concept of right-upper derivative and the concept of sub linear. The sufficient and necessary conditions of the static programming were d erived in terms of a generalized Lemma in this paper. The results obtained are u seful for the further study on the duality of static programming and cover many already known conditions.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.
基金National Natural Science Foundation ofChina(No.70273029)
文摘The definitions of generalized pseudoconvex,generalized quasiconvex and its stri ctly generalized convexity were presented for the static programming at locally star -shaped set using the concept of right-upper derivative and the concept of sub linear. The sufficient and necessary conditions of the static programming were d erived in terms of a generalized Lemma in this paper. The results obtained are u seful for the further study on the duality of static programming and cover many already known conditions.
