So, clearly, starting at , than following the cycle infinitely many times, the DFA can accept infinitely many strings.

Hence, L(M) is infinite.

It is a well-known fact that there exists efficient algorithm to detect a cycle in directed graph. From the above, we have the following efficient algorithm to decide infiniteness of L(M).

Delete all states not reachable from the start state and delete all states that cannot lead to an accept state ( DFS can be used for this). If there is a cycle, then L(M) is infinite. Otherwise, L(M) is finite.