关于图运算的测地数

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SV(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狖g(D):D是G的定向图狚,G的上测地数g+(G)=max狖g(D):D是G的定向图狚.对于两个图G和H,u∈V(G)和v∈V(H),在u和v之间加一条边,然后再收缩这条边uv所得的图,记为GuHv.本文主要研究图GuHv的测地数和上(下)测地数.

The Geodetic Number on a Graphic Operation

For any two vertices u and v in a graph G (or digraph D), a u- v geodesic is a shortest path between u and v (or from u to 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) (or g(D)) of a graph G (or digraph D) is the minimum cardinality of a set S with I(S)=V(G) (or I(S)=V(D)).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),G joining H at u and v, denoted by Gu o Hv, is defined as follows: adding an edge between u and v, then contracting the edge uv. The main purpose of this paper is to study the lower and upper geodetic numbers of the graph Gu o Hv.