Let's pretend that we do a DFS on a directed graph Gd, where G is the depth-first tree or forest corresponding to Gd. There will be no cycles in the final graph G if we eliminate all the back edges with regard to Gd.    What do you think?

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Minimum Spanning Tree

A spanning tree is a sub-graph of a connected, undirected graph. It has all the vertices of a graph with the minimum possible number of edges. If a vertex is skipped, then it will not be a spanning tree. The edges of a spanning tree may or may not have weights.