图论知识点总结
一. 子图
（1）spanning subgraph(生成子图):Let G = (V, E) be a graph and H = (V’, E’) be a subgraph of G.The subgraph H is a spanning subgraph of G if V’ = V
（2）edge induced subgraph(边导出图):Let G = (V, E) be a graph. If E' belong to E, the subgraph of G induced by E’ is the graph H = (V’, E’) where v belong to V’ if and only if v appears in an edge in E’.
（3）vertex induced subgraph（点导出图）:Let G = (V, E) be a graph. If V’ belong to V, V’ is not a empty set , The subgraph of G induced by V’ is the graph H = (V’, E’) where {v1, v2} belong to E’ if and only if v1 and v2 are both in V’.
二. regular graph(正则图)
defination: Let G = (V, E) be a graph with |V| = n. If the degree sequence of G is (k, k, …, k) with k <= n-1 then G is called a kregular graph on n vertices.
三. 团(clique)
defination: 对于给定图G，其团就是一个两两之间有边的顶点集合。
(1)maximal clique：不属于任何团的真子集的团
(2)maxinum clique: 顶点数最大的极大团
(3)clique number: ω(G) = 最大团的顶点数
四. 独立集（independent set）
defination:两两互不相邻的顶点组成的集合
(1)independence number: α（G） = 最大独立集的顶点数
五. vertex covering
defination: Let G = (V, E) be a graph. A vertex cover is a set of vertices C belong to V so that for any e belong to E at least one element of e is in C; i.e.,every edge in E is adjacent to at least one vertex in C.
六. 补图
defination： 一个图G的补图（complement）或者反面（inverse）是一个图有着跟G相同的点，而且这些点之间有边相连当且仅当在G里面他们没有边相连
(1)Relative Complement：If G = (V, E) is a graph and H = (V, E’) is a spanning sub-graph of G, then the relative complement of H in G is the graph H’= (V, E”) with:
for any v1, v2 belong toV, e = {v1, v2} belong to E” iff {v1, v2} belong toE and {v1, v2}is not belonging to E’
七. 覆盖集和独立集的关系
(1) Let G = (V, E) be a graph with n vertices and there is no isolated vertex in G. a and b are used to present vertex covering number and vertex independent number respectively. Then
a + b = n
a + b = n
(2)Let G = (V, E) be a graph and there is no isolated vertex in G. A set C belong to V is a vertex covering in G if and only if the set V-C is an independent set in G.