In a depth first tree, there are the edges define the tree (i.e the edges that were used in the traversal).
There are some leftover edges connecting some of the other nodes. What is the difference between a cross edge and a forward edge?
Based on this spanning tree, the edges of the original graph can be divided into three classes: forward edges, which point from a node of the tree to one of its descendants, back edges, which point from a node to one of its ancestors, and cross edges, which do neither. Sometimes tree edges, edges which belong to the spanning tree itself, are classified separately from forward edges. If the original graph is undirected then all of its edges are tree edges or back edges.
Doesn't an edge that is not used in the traversal that points from one node to another establish a parent-child relationship?