# A graph which is not a single block has at least two leaf blocks

A graph which is not a single block has at least two leaf blocks.

According to the Intro. to Graph Theory by D. West:

• Block: A maximal connected subgraph of $$G$$ that has no cut-vertex.
• Leaf block: A block that contains eactly one cut-vertex of $$G$$.

I have not well understood this statement. Would you elaborate on this statement.

Edit: Can we say this:

The block-cutpoint graph of a connected graph is a tree where its leaves denote the blocks of $$G$$. Since a tree has at least two leaf blocks hence the graph has at least two leaf blocks if it is not a single block.

• What is a "block"? – Henning Makholm Mar 24 at 10:39
• @HenningMakholm I just edited the question. – Amin Mar 24 at 10:44
• Sorry, if you take 2 disjoint $K_3$ as your graph $G$, then $G$ is not a single block but neither of the 2 $K_3$ are leaf blocks? – user 42493 Mar 24 at 10:56
• @user42493: we assume it is connected. – Amin Mar 24 at 11:22