I am trying to find in the following graph the strongly connected components but i have some questions since in the class we picked up on the topic very briefly. Are loops considered in this example strongly connected components?Also, if a strongly connected component includes a vertex can this vertex be included in another strongly connected component?
The strongly connected components i think are in this graph are:
sub-graph induced by {e}
sub-graph induced by {a,d}
sub-graph induced by {a}
sub-graph induced by {f}
sub-graph induced by {b,c,e}
Can you guide me through the logic?
Looks like your main struggle is with the definitions; we have that a directed graph (digraph) is strongly connected exactly when there is a directed path between any pair of vertices. A strongly connected component is a maximal strongly connected subgraph (note that in this context we can safely ignore loops, since they don't impact strong connectivity). Call the digraph in question $D$; this is to say that vertices of the strongly connected components of $D$ partition $V(D)$. This immediately voids the subgraph $D[a]$ since, as you've found, it is not maximal (the subgraph induced by {a, d} (i.e. $D[a, d]$) contains it). In light of this, it is simple to see that the strongly connected components are $D[a, d]$, $D[b, c, e]$, and $D[f]$. You'll need to confirm for yourself that all of these are maximal; I'll give the example argument for $D[f]$: Since the only in-neighbor of the vertex $f$ is $d$, and there is no directed path from $f$ to $d$ in $D$, this strongly connected component cannot be any larger -- hence it is maximal.