Connectivity and decomposition into connected components

A graph is connected if for every pair of vertices $u$ and $v$, there exists a path from $u$ to $v$. If a graph is not connected, it can be uniquely partitioned into maximal connected subgraphs, called connected components. Each component is connected, and there are no edges between different components.

Important properties

• Every vertex belongs to exactly one connected component.

• A connected graph has exactly one connected component (itself).

• The connected components of a graph partition its vertex set.

• Removing all edges between components does not change the components.