I'm reading Graph Theory from Kenneth H. Rosen.
A connected component of a graph G is a connected subgraph of G that is not a proper subgraph of another connected subgraph of G. That is, a connected component of a graph G is a maximal connected subgraph of G.
This confuses me, as I can see from the example that H3 (connected component) is a subgraph of H1 (connected component).
What are the connected components of the graph H shown in Figure 3?
A subgraph of $H_1$ is a graph that shares all of its points and edges with $H_1$. For example, the graph consisting of points $abc$ and edges $ab, bc$ is a subgraph of $H_1$.
$H_3$ shares no points or edges with $H_1$, so it is not a subgraph of it. It is isomorphic to a subgraph of $H_1$, but that's not the same as actually being a subgraph of $H_1$.