I understand that a graph is connected if there is a path between any two vertices.
However, I am unsure about the number of components of a graph. Is a single vertex considered a component? If yes, is it connected?
In particular, what is the number of components in the below graph? Is it one or three?
By definition, for an undirected graph, i.e. a graph where edges have no direction associated with them a (connected) component is basically a set of vertices that are inter-reachable, or also a connected subgraph, and since isolated vertices can only "reach" themselves they are to be considered as connected components. Note that the word connected may be omitted but any component is still to be considered connected since this concept is intrinsic to the definition. Hence each "component" and "connected component" are typically intended as synonyms.
It follows that in your example there are $3$ connected components: $\left\{i\right\}$, $\left\{f\right\}$ and $\left\{e,g,h\right\}$.
In the case of directed graphs instead, we do not talk about connected components, but we distinguish between strongly and weakly connected components.
