Union-Find algorithm and determining whether an edge belongs to a cycle in a graph

Question

I'm reading a book about algorithms ("Data Structures and Algorithms in C++") and have come across the following exercise:

Ex. 20. Modify cycleDetectionDFS() so that it could determine whether a particular edge is part of a cycle in an undirected graph.

In the chapter about graphs, the book reads:

Let us recall from a preceding section that depth-first search guaranteed generating a spanning tree in which no elements of edges used by depthFirstSearch() led to a cycle with other element of edges. This was due to the fact that if vertices v and u belonged to edges, then the edge(vu) was disregarded by depthFirstSearch(). A problem arises when depthFirstSearch() is modified so that it can detect whether a specific edge(vu) is part of a cycle (see Exercise 20). Should such a modified depth-first search be applied to each edge separately, then the total run would be O(E(E+V)), which could turn into O(V^4) for dense graphs. Hence, a better method needs to be found.

The task is to determine if two vertices are in the same set. Two operations are needed to implement this task: finding the set to which a vertex v belongs and uniting two sets into one if vertex v belongs to one of them and w to another. This is known as the union-find problem.

Later on, author describes how to merge two sets into one in case an edge passed to the function union(edge e) connects vertices in distinct sets.

However, still I don't know how to quickly check whether an edge is part of a cycle. Could someone give me a rough explanation of such algorithm which is related to the aforementioned union-find problem?

Answer 1

a rough explanation could be checking if a link is a backlink, whenever you have a backlink you have a loop, and whenever you have a loop you have a backlink (that is true for directed and undirected graphs).

A backlink is an edge that points from a descendant to a parent, you should know that when traversing a graph with a DFS algorithm you build a forest, and a parent is a node that is marked finished later in the traversal.

I gave you some pointers to where to look, let me know if that helps you clarify your problems.