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

We can visualize a DFS traversal by orienting the edges along the direction in
which they are explored during the traversal, distinguishing the edges used to
discover new vertices, called discovery edges, or tree edges, from those that lead
to already visited vertices, called back edges. (See Figure 13.6f). In the analogy
above, discovery edges are the edges where we unroll our string when we traverse
them, and back edges are the edges where we immediately return without
unrolling any string. As we will see, the discovery edges form a spanning tree of
the connected component of the starting vertex s. We call the edges not in this tree
"back edges" because, assuming that the tree is rooted at the start vertex, each
such edge leads back from a vertex in this tree to one of its ancestors in the tree.
The pseudo-code for a DFS traversal starting at a vertex v follows our analogy
with string and paint. We use recursion to implement the string analogy, and we
assume that we have a mechanism (the paint analogy) to determine if a vertex or
edge has been explored or not, and to label the edges as discovery edges or back
edges. This mechanism will require additional space and may affect the running
time of the algorithm. A pseudo-code description of the recursive DFS algorithm
is given in Code Fragment 13.1.
Code Fragment 13.1: The DFS algorithm.
There are a number of observations that we can make about the depth-first search
algorithm, many of which derive from the way the DFS algorithm partitions the
edges of the undirected graph G into two groups, the discovery edges and the
back edges. For example, since back edges always connect a vertex v to a
previously visited vertex u, each back edge implies a cycle in G, consisting of the
discovery edges from u to v plus the back edge (u, v).
Proposition 13.12: Let G be an undirected graph on which a DFS
traversal starting at a vertex s has been performed. Then the traversal visits all
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