Finding all cycles that do not contain sub-cycles in an undirected graph

Question

I'm trying to find a fast way to find all the cycles that do not contain any sub-cycles within them in an undirected graph.

Here's an example of a graph that contains five cycles of this kind:

The cycles here would be the following: {1,2,3,4}, {1,4,5}, {5,4,7,6}, {4,3,7}, {7,3,9,8}

How could I approach this problem?

Answer 1

Cycles that don't contain subcycles are called " induced cycles ", if you search this term maybe you will find something that you will find of use.

For example, I found this article regarding exactly the problem you have described: Finding and listing induced paths and cycles .

For my personal needs I recently programmed an algorithm that finds cycles without subcycles in a directed graph, but it will require some modifications to work for undirected graphs, also I'm not sure how fast will it be if used on an undirected graph.