SKEDSOFT

Introduction: A vertex v of a graph G is a cut vertex or an articulation vertex of G if the graph G−v consists of a greater number of components than G.
Example. v is a cut vertex of the graph below:

(Note! Generally, the only vertex of a trivial graph is not a cut vertex, neither is an isolated vertex.)
A graph is separable if it is not connected or if there exists at least one cut vertex in the graph. Otherwise, the graph is nonseparable.

Example. The graph below is nonseparable.

A block of the graph G is a subgraph G1 of G (not a null graph) such that
• G1 is nonseparable, and
• if G2 is any other subgraph of G, then G1 ∪G2 = G1 or G1 ∪G2 is separable (think about that!).

Example. The graph below is separable: