If the graph is directed, do an SCC decomposition in linear time using a graph library and then any SCC with size more than one has at least a loop, trivially extracted by following any edges in the SCC.
If the graph is undirected, compute a spanning tree in linear time using a graph library and then any edge outside the tree forms a loop, again trivially extractable from the edge and the paths from its endpoints to their least common ancestor in the tree.
Since those algorithms are already asymptotically optimal, it's just an engineering problem of finding the solution with the best constant factor given the precise data structures and goal in question.
The most obvious way is to just run a DFS, if you ever come back to a node you’ve seen then you have a cycle (and by storing parent pointers you can get the cycle back). That’s like CS 101 level.
However in the article it’s actually a graph being modified. If you don’t want to run a full dfs every time, there are better ways. This problem is called “online cycle detection” and you can google some other algorithms.
You obviously didn’t read the article. It’s about undirected graphs, and about visualizing the loops, not just testing for their existence. The final solution is indeed based on spanning trees.
If the graph is directed, do an SCC decomposition in linear time using a graph library and then any SCC with size more than one has at least a loop, trivially extracted by following any edges in the SCC.
If the graph is undirected, compute a spanning tree in linear time using a graph library and then any edge outside the tree forms a loop, again trivially extractable from the edge and the paths from its endpoints to their least common ancestor in the tree.
Since those algorithms are already asymptotically optimal, it's just an engineering problem of finding the solution with the best constant factor given the precise data structures and goal in question.