Tree Edge, front edge, back edge, cross side

Original address: Tree Edge, front edge, back edge, cross fork Edge

Tree edge, forward edge, back edge, cross side, it should be said, is not a concept of the diagram itself, should be the concept of Dfs. The figure for DFS will get a Dfs tree (forest) that has these concepts on this tree. DFS, can start from any vertex, the way of traversal is also diverse, so different traversal will get different Dfs tree, resulting in different tree edge, forward edge, back edge, cross edge. So these 4 kinds of sides, is a relative concept.
In the traversal of graphs, a bool value of a tag array vis is often set to record whether the vertices have been accessed. But sometimes it is necessary to change the meaning of the VIS value. The VIS has 3 values and represents 3 different meanings
Vis = 0, indicating that the vertex has not been accessed
Vis = 1, indicating that the vertex has been accessed, but its descendants have not yet been visited, and have not returned from that point
Vis = 2, indicating that the vertex has been accessed and that its descendants have already visited it, and has returned from that vertex
The 3 values that can be vis represent a sequence relationship and a time relationship

"Introduction to Algorithms" 334 pages have the exact definition of these 4 kinds of edges, not in this statement
DFS process, for an edge u->v
VIS[V] = 0, indicating v has not been accessed, V is the first to be discovered, U->v is a tree edge
VIS[V] = 1, indicating that V has been visited, but its descendants have not yet been visited (in the access), and you point to V? Note that u is a descendant of V, U->v is a back edge, so the back edge is also called the atavistic edge
VIS[V] = 3,z indicates that V has been visited and that its descendants have all been visited, u->v the edge may be a cross-border, or forward edge

Note: Tree-side, back-side, forward-side, all have ancestors, descendants of the relationship, but the cross-side is not, u->v for the cross-side, indicating that in this Dfs tree, they are not ancestor descendants of the relationship they may be brotherly relations, cousin relationship, or even farther relationship, if it is the Dfs forest, U and V can even be on different trees

In many algorithms, the back edge is useful, but the effects of the forward and cross edges are often diluted, but they do not have much effect.

The previous picture below:

Tree Edge
Cross Edge with fork edge
Forward Edge Forward Edge
Rear Edge back Edge

Tree Edge, front edge, back edge, cross side