Connected Components

What if your graph is not one connected piece?

Imagine islands in an ocean. Bridges connect some islands, but not all. Each cluster of connected islands is a component. In graph terms, a connected component is a maximal set of vertices where every pair is reachable from each other. If you have vertices $\{1, 2, 3, 4\}$ with edges $1-2$ and $3-4$, there are two components: $\{1, 2\}$ and $\{3, 4\}$. Counting components is a common problem. DFS makes it easy.