树的笛卡儿积的测地数

The Geodetic Numbers of Cartesian Products of Trees

图G内的任意两点u和u,u-v测地线是指u和v之间的最短路.I(u,v)表示位于u-v测地线上所有点的集合,对于子集S(?)V(G),I(S)表示所有I(u,v)的并,这里u,v∈S.图G的测地数g(G)是使I(S)=V(G)的点集S的最小基数.本文研究了任意连通图G与树T笛卡儿积的测地数的界,同时,给出了任意两个树T^1与T^2笛卡儿积的测地数和树T与圈C笛卡儿积的测地数.

For any two vertices u and v in a graph G, a u- v geodesic is the shortest path between u and v. Let I（u, v） denote the set of all vertices lying on a u - v geodesic. For a vertex subset S, let I（S） denote the union of all I（u, v） for u, v ∈ S. The geodetic number g（G） of a graph G is the minimum cardinality of a set S with I（S） = V（G）. In this paper, we give some bounds of g（G × T） for any graph G, where T is a tree. Moreover, the geodetic number of T^1 × T^2 and Ck × T are presented, where T^1 and T^2 are trees, Ck is a cycle of order k.