The cycle detection problem for a directed graph has well-known polynomial time

Question

The cycle detection problem for a directed graph has well-known polynomial time solutions, graph traversal algorithms such as Dijkstra algorithm can be used to find whether or not a cycle exists in a graph.

The difficulty however comes when we are required to find in polynomial time whether 2 vertex disjoint cycles exist in an arbitrary graph. Is there such a polynomial time algorithm which finds whether 2 vertex disjoint cycles exist in an arbitrary graph?

By vertex disjoint cycles, I mean cycles which do not share a common vertex.

Explanation / Answer

The structure of graphs without two vertex-disjoint cycles is well understood (see for example Bollob

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