Some Graph's Geodetic Set

The terminology of geodesic originated from the convex set theory of geom-etry, topology and functional analysis and have very important applications and significance on Site selection, Network design, Control theory and so on. After in-troducing convexity to graph theory, geodesics problems and the related measure i.e. geodetic number also prove to be an important index and parameters for reveal-ing the graph structural.This thesis mainly investigates geodetic sets and some other geodetic sets with special properties of graphs, such as edge geodetic sets, linear geodetic sets. Be-sides, the edge(or linear) geodetic number of some graphic operations are charac-terized.(i) In Chapter 1, we introduced simply the history of graph theory,the back-ground of geodetic sets(geodetic number) and research progresses. Besides, some definitions and notations for the corresponding questions are given.(ii) Chapter 2 of the first section, We first studied the properties of geodetic sets of graphs surveying set properties and revealed the relation between geodetic sets and other structural parameters of graphs. On this basis, geodetic numbers of Unicycle graphs and complete r partite graphs are obtained.In second section, geodetic number of some class graphs are obtained. Mean-while, the relationship of geodetic number and edge geodetic number are character-ized. Here is the main results of this section:(1) Let G be a tree Tn(or complete graphs Kn, Kn-e, Km,n, Km1,…,mr and Unicycle graph Gn). then g(G)=ge(G). (2) For any positive integer 2≤a≤b, there must be a graph G such that: g(G)= a,ge(G)=b.In third section, we studied linear geodetic sets and linear geodetic number, the corresponding indexes of some class of graphs are characterized. The main results of this section as follows:(1) Let T be a tree of k leafs. then gl(T)=k, and for any labels of vertexes of leafs set:S={v1,v2,…,vk}, S must be a smallest linear geodetic set of T.(2) Let G(?) Cn be a Unicyclic graph and all trees planted on cycle is not paths. Then gl(G)=g(G).(iii) In Chapter 3, We considered two graphic operations and its influence on edge(or linear) geodetic number. The results as follows:Let S1,S2 are two smallest edge(or linear) geodetic sets of nontrivial graphs G, H, respectively. If u∈Si,v∈S2, then(1) ge(Gu+Hv)=ge(G)+ge(H)-2, gl(Gu+Hv)=g,(G)+gl(H)-2.(2) ge(Gu o Hv)=ge(G)+ge(H)-2, gl(Gu(?)Hv)=gl(G)+gl(H)-2.