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based on an earlier version of the Wolfram Language.
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DepthFirstTraversal[g, v]
performs a depth-first traversal of graph g starting from vertex v, and gives a list of vertices in the order in which they were encountered.
DepthFirstTraversal[g, v, Edge]
returns the edges of the graph that are traversed by the depth-first traversal in the order in which they are traversed.
DepthFirstTraversal[g, v, Tree]
returns the depth-first tree of the graph.