I read the following statement :
If the graphs $G$ and $G'$ are isomorphic then following is true:
- If $G$ is connected, so is $G'$. More generally, $G$ and $G' $ have same number of connected components.
I don't understand what connected components mean. Can anyone explain this to me, please?
Define the relation "is reachable", denoted $ \sim $, on the set $V \times V$, where $V$ is the set of vertices of the given graph $G$, by ${v_1} \sim {v_2} \Leftrightarrow $ there exists a path from ${v_1}$ to ${v_2}$ in $G$ which begins in ${v_1}$ and ends in ${v_2}$ or ${v_1} = {v_2}$.
Assuming that $G$ is an undirected graph, check for yourself that the $ \sim $ is an equivalence relation.
As you probably know, all equivalence relations induce a partition on the given set (here $V$) into equivalence classes. Connected components are the equivalence classes induced by $ \sim $.