Question

I have a directed cyclic graph. Some edges are FIXED and may not be removed. The other edges may be removed to break a cycle.

What is the best way to do remove the cycles in this graph?
The traversal should be as much as a DFS as possible and start at a given node.

Answer 1

I used the following algorithm to solve my problem:

Start with a graph of all fixed edges marked as confirmed

From a start node, recurse through all confirmed edges as well as the not-yet-confirmed ones. But when you’re about to recurse down a not-yet-confirmed edge first check if the node that the edge goes to has a path, by following confirmed edges, to a node in the current search tree (i.e. a node with a visiting flag set). This check must be done recursively by following all confirmed edges but this is too slow for me so I will settle here to just check if the node is visiting or if any of the nodes it is confirmed connected to are visiting. This will cover most of my cases but occationally leave cycles in the graph.

After the above step I use Tarjan’s algorithm for finding the strongly connected components left in the graph (this can be done in O(V + E) time). The number of strongly connected components will be very small in my cases so I just go through each strongly connected component and remove one removable edge each. Then I do this step again until no more cycles remain in the graph.

This works fine and is fast enough.