I am competing in the American Computer Science League (ACSL), and I get problems similar to the following.
Look at this directed graph. Now tell me the number of cycles in said graph.
These questions need to be completed quickly and by hand. Unfortunately, the ACSL wiki (which provides help and study resources for the test) says to find all cycles by inspection. This obviously is slow and possible erroneous. I want to be able to complete these questions with the utmost accuracy. Is there a method to, given a graph, calculate the number of cycles by hand?
There are algorithms that would do this in general, but they are overkill for such a small graph.
My strategy (assuming we want to find/count elementary cycles) for this particular problem would be to first note the 2-cycles ADA and BDB, and then draw 4 graphs for the 4 possible choices of direction on AD and BD.
Out of those 4 graphs, one can immediately be discarded because B will have only incoming edges and D will have only outgoing edges. In another, B can be deleted because it has only incoming edges. This leaves two graphs between which the only difference is the direction of AD.