Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

A DFS on a digraph partitions the edges of reachable from the starting
vertex into tree edges or discovery edges, which lead us to discover a new vertex,
and nontree edges, which take us to a previously visited vertex. The tree edges
form a tree rooted at the starting vertex, called the depth-first search tree, and there
are three kinds of nontree edges:
• back edges, which connect a vertex to an ancestor in the DFS tree
• forward edges, which connect a vertex to a descendent in the DFS tree
• cross edges, which connect a vertex to a vertex that is neither its ancestor
nor its descendent.
Refer back to Figure 13.9b to see an example of each type of nontree edge.
Proposition 13.16: Let be a digraph. Depth-first search on starting at
a vertex s visits all the vertices of that are reachable from s. Also, the DFS tree
contains directed paths from s to every vertex reachable from s.
Justification: Let V s be the subset of vertices of visited by DFS starting
at vertex s. We want to show that V s contains s and every vertex reachable from s
belongs to V s . Suppose now, for the sake of a contradiction, that there is a vertex w
reachable from s that is not in V s . Consider a directed path from s to w, and let (u, v)
be the first edge on such a path taking us out of V s , that is, u is in V s but v is not in
V s . When DFS reaches u, it explores all the outgoing edges of u, and thus must
reach also vertex v via edge (u,v). Hence, v should be in V s , and we have obtained a
contradiction. Therefore, V s must contain every vertex reachable from s
Analyzing the running time of the directed DFS method is analogous to that for its
undirected counterpart. In particular, a recursive call is made for each vertex exactly
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