图的下测地数和上测地数

对于图G(或有向图D)内的任意两点u和v,u—v测地线是指在u和v之间(或从u到v)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S(?)V(G)(或V(D)),I(S)表示所有I(u,v)的并,这里u,v∈S．G(或D)的测地数g(G)(或g(D))是使I(S)=V(G)(或I(S)=V(D))的点集S的最小基数．G的下测地数g~-(G)=min{g(D)：D是G的定向图},G的上测地数g~ (G)=max{g(D)：D是G的定向图}．对于u∈V(G)和v∈V(H),G_u H_v表示在u和v之间加一条边所得的图．本文主要研究图G_u H_v的测地数和上(下)测地数．

The Lower and Upper Geodetic Numbers of Graphs

For any two vertices u and v in a graph G (digraph D, respectively), a u- v geodesic is a shortest path between u and v (from u to v, respectively). 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) (g(D), respectively) of a graph G (digraph D, respectively) is the minimum cardinality of a set S with I(S) = V(G)(I(S) = V(D), respectively). The lower geodetic number of G is g-(G) = min{g(D): D is an orientation of G}. The upper geodetic number of G is g+(G) = max{g(D): D is an orientation of G}. For two graphs G and H with u ∈ V(G) and v ∈ V(H), Gu + Hv is a graph obtained from G and H by adding an edge uv. The main purpose of this paper is to study the lower and upper geodetic numbers of the graph Gu + Hv.