# Decidability of directed strongly connected graphs

Consider the problem of determining if a directed graph is strongly connected.
How to phrase it as a language and prove that it's decidable.

My Thoughts :
To think of decidability given a graph I could run DFS on each node and see that all the nodes
are reachable from the original node. So it can be determined if the graph is strongly connected
in polynomial time.
But I am confused about the formal automata definition of decidability. Which says a language is
decidable if there is a Turing machine such that this language is the one exactly accepted by the
Turing machine. Now given the above logic of DFS is that enough to prove decidability
or does one have to build a turing machine ?

Btw, as a side note, you can run DFS only twice in total instead of $$n$$ times - once per node. If you have free time, its a good exercise in algorithms :)