[Connectivity] FIG.
1 , a loop: the same start and end points, length is greater than 0 .
It is not necessary to distinguish between multiple edges, the sequence may be expressed via respective vertex.
A length of 0 passage of a single vertices.
Simple passages: Unique edge
Primary path: Unique point (path)
(Tip Memory: "primary", and therefore, a smaller range than the primary "simple" lower, etc.)
2, with the patterned invariants: length k existence of loop √
3, undirected graph connectivity: if any two vertices are in communication, called the no-communication to FIG.
Communicating branches: "path exists between the vertex" is an equivalence relation on either induced subgraph equivalence class that is a branch of a communication.
. 4, (⭐) cut point: Remove this point will increase the number of communicating branches (likewise defined cutting edge)
A major cutting edge on Theorem: e is the cutting edge if and only if e is not G any of a simple loop (proved relatively simple, abbreviated)
(Note: The cutting edge is clearly stronger than the cut point, because the cutting edge just delete an edge in the figure, while the cut point of at least two edges to be deleted.)
5, the communication of FIG.
Definition: non-trivial connected graph G becomes minimum number of vertexes or not trivial FIG communication to be deleted is referred to FIG. FIG G 's ( point ) connectivity, referred to as [kappa] (G)
(Knock on the blackboard: "at least", indicating that connectivity is just an optimal situation, for any k vertices not be true )
Conventions: no connectivity or connectivity graph of FIG ordinary 0 , and [kappa] (Kn) =. 1-n- , if FIG G connectivity is not less than k, called G is k- connected graph
6, the upper limit of the communication (⭐): FIG If G is nontrivial , then k (G) <= lamda ( G) <= G smallest degree.
(Rear inequality is clear that for the inequality front, it is conceivable, by deleting a vertex can be deleted at least one side. Therefore, by removing the cost side to be larger, so the inequality holds. Or put it another thought, you can All the cutting edge and its corresponding vertex to gather a set of pattern is not necessarily outside the set of communication, and for the collection of internal cost, is clearly higher side deleted. Thus, Lamda (G)> = K (G) )
[Ps: FIG communication reaches the upper limit]
7 , Whitney theorem:
FIG G (| G |> = 3 ) is 2- connected graph, if and only if G any two points in the primary circuit are at the same
Menger Theorem ( Whitney promotion Theorem)
- Fig G is a k- communicate if and only if G any two points is at least k connected paths except end vertex disjoint.
- Fig G is a k- edge connected graph if and only if G any two points is at least k paths intersecting edges not connected.
8 , two connected graph ( H-path that part yet get to know ... I need to be careful I read about it, they get to know it will update ... )
9, there is communication to FIG.
Communication weak / strong communication
Strong determined communication Theorem: digraph D when it is strongly connected if and only if D all vertices in a directed in the same loop.
To be continued ... (will continue modified)