Connectivity, Cut Edge, and Cut Vertex
==============================

Connectivity
-------
A graph 
G
is **connected** if for every pair of vertices 
u
and 
v, there exists a path from 
u to 
v.
Otherwise, it is **disconnected**. Every disconnected graph is a collection of several connected graphs, each of which is called a **component**. 
Clearly, a connected graph has exactly one component.

If we imagine vertices as balls and edges as strings, the components of the graph can be 
physically separated by hand, but the vertices within each component remain connected by strings.

Cut Vertex 
----------
In graph 
G, a vertex 
v 
is called a **cut vertex** if removing it increases the number of connected components.

Cut Edge 
----------
In graph 
G, an edge 
e is called a **cut edge** if removing it increases the number of connected components.

k-Connected Graph
----------------

A graph 
G
is called **k-connected** if it has more than 
k vertices and cannot be disconnected by removing 
x 
vertices (where x < k). 
The **connectivity** of G, denoted as 
:math:`\kappa (G)`, is defined as the maximum 
k for which 
G is 
k-connected. In other words, it is the minimum number of vertices whose removal disconnects the graph or reduces it to a single vertex.

Block
-----
A **block** is a maximal subgraph of 
G that contains no cut vertices.