Purify the cycles in undirected graphs

[YusufZiyaOzgul]

For Paths Between query, there is a cycle problem. if graph is undirected.
For example,

In this example, the only path not including cycle is the path( shown red ) from glyph8 to glyph17. However, algorithm also finds the paths which inludes cycles. For example, glyph17-glyph10-glyph17 is a path includes cycle.

To eliminate this problem, we need to purify the graph after we find the paths. Purify operation is mentioned in this article but details of it is not given in article. Purify operation should be implemented to solve this problem.