DepthFirstScan
DepthFirstScan[g,s,{"event_{1}"f_{1},"event_{2}"f_{2},…}]
performs a depthfirst scan of the graph g starting at the vertex s and evaluates f_{i} whenever "event_{i}" occurs.
DepthFirstScan[g,{"event_{1}"f_{1},"event_{2}"f_{2},…}]
performs a depthfirst scan of the whole graph g.
DepthFirstScan[{vw,…},…]
uses rules vw to specify the graph g.
Details
 DepthFirstScan[g,s,{…}] visits vertices in the graph g connected to the vertex s in depthfirst order.
 In depthfirst order, vertices adjacent to the most recently visited are visited first.
 DepthFirstScan[g,{…}] performs multiple depthfirst scans starting from the first vertex in the vertex list of g, then starts from the first vertex in the vertex list that has not been visited, and so on, effectively scanning each connected component.
 DepthFirstScan[g,…] gives a list {w_{1},w_{2},…} representing a tree where w_{i} is the predecessor of v_{i} and where {v_{1},v_{2},…} is the vertex list of g.
 Events that provide access to vertex discovery include:

"DiscoverVertex" when vertices are discovered "UnvisitedVertex" when unvisited vertices are rediscovered "VisitedVertex" when visited vertices are rediscovered  "DiscoverVertex">fd calls fd[u,v,d] when vertex u is discovered from visited vertex v at distance d from the start vertex s.
 "UnvisitedVertex">fru calls fru[u,v] when the unvisited vertex u is rediscovered from the visited vertex v.
 "VisitedVertex">frv calls frv[u,v] when the visited vertex u is rediscovered from the visited vertex v.
 Events that provide access to vertex visits include:

"PrevisitVertex" before a vertex is visited "PostvisitVertex" after a vertex has been visited  "PrevisitVertex">fs calls fs[u] before the vertex u has been visited.
 "PostvisitVertex">fe calls fe[u] after the vertex u has been visited.
 The DFS tree is the tree generated by the edges traversed during a depthfirst scan.
 Events that provide access to edge exploration from the visited vertex include:

"FrontierEdge" edge in the DFS tree "BackEdge" edge to ancestor in the DFS tree "ForwardEdge" edge to descendant in the DFS tree "CrossEdge" other edge  "FrontierEdge">ffe calls ffe[vu] for an edge vu where the vertex v is being visited and u has not been discovered. This is typically useful for scanning the DFS tree.
 "BackEdge">fbe calls fbe[vu] for an edge vu where the vertex v is being visited and u has already been discovered and is an ancestor to v in the DFS tree. This is typically useful for finding loops.
 "ForwardEdge">gfe calls gfe[vu] for an edge vu where vertex v is being visited and u has already been discovered and is a descendant to v in the DFS tree.
 "CrossEdge">fce calls fce[vu] for an edge vu where v is being visited and u has already been discovered and is not in the current DFS tree or is in the same DFS tree, but in a different branch. This is typically useful for detecting multiple DFS trees.
 For an undirected graph, the edges used in the callbacks are taken to be undirected edges vu.
Background & Context
 DepthFirstScan performs a traversal of a graph starting at a root vertex and exploring as far as possible along each branch before backtracking. At each relevant event during the scan (vertex discovery, unvisited vertex discovery, or vertex rediscovery), a userdefined function may be evaluated. Depthfirst traversal is useful for computing many graph properties since, as shown by Tarjan and Hopcroft in the early 1970s, it results in lineartime algorithms for many problems in graph theory. Using DepthFirstScan therefore allows custom userdefined graph theoretical algorithms to be implemented efficiently. Example applications include finding connected components, finding bridges, planarity testing, and topological sorting.
 In depthfirst order, vertices adjacent to the most recently visited are visited first. A depthfirst scan may be used to search or visit all vertices and edges of a graph, or only those starting at a specified vertex.
 A depthfirst scan is sometimes also known as a depthfirst search (DFS) or depthfirst traversal. The French mathematician Charles Pierre Trémaux first studied a version of depthfirst scanning in the 19th century as a way to solve mazes.
 BreadthFirstScan is a similar traversal algorithm that begins at a root vertex, inspects all the neighboring vertices, visits the neighbors of the neighbors it just inspected, and so forth.
Examples
Text
Wolfram Research (2010), DepthFirstScan, Wolfram Language function, https://reference.wolfram.com/language/ref/DepthFirstScan.html (updated 2015).
CMS
Wolfram Language. 2010. "DepthFirstScan." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/DepthFirstScan.html.
APA
Wolfram Language. (2010). DepthFirstScan. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DepthFirstScan.html