Analyzing Depth-first Search Edge Types

[evinda]

Hello! (Wave)

I want to apply the Depth-first search algorithm at the following graph.
We consider that at the iteration of the algorithm, we look at the nodes alphabetically. Also the nodes are registered alphabetically in each list. I want to calculate the "discovery" time and the "finish" time for each node and also the kind of each edge.

View attachment 3654

Depthfirstsearch(G)
   for each v ∈ V
        color[v]=white
        p[v]=NIL
   time=0
   for each v ∈ V
       if color[v]=white then
          Visit(v)

Visit(u)
  color[u]=gray
  time=time+1
  d[u]=time
  for each v ∈ Adj[u]
       if color[v]=white then
          p[v]=u
          Visit(v)
  color[u]=black
  time=time+1
  f[u]=time

I found these "discovery" and "finish" times:

View attachment 3655

Have I calculated them right? (Thinking)

Also, in order to find the kind of the edges I wanted to use this:

$$\begin{bmatrix}
\text{ tree edges: } x \to y & [d[y],f[y]] \subset [d[x],f[x]] \\ \\
\text{forward edges: } x \to y & [d[x],f[x]] \subset [d[y],f[y]] \\ \\
\text{back edges: } x \to y & [d[y],f[y]] \subset [d[x],f[x]] \\ \\
\text{Cross edges: } x \to y & [d[x],f[x]] \cap [d[y],f[y]]=\varnothing
\end{bmatrix}$$

But, when we have for example the case $[d[y],f[y]] \subset [d[x],f[x]]$ how can we know if it is a tree edge or a back edge? (Thinking)

 

[evinda]

I think that I have done some mistakes.. (Lipssealed)
I think that it should be like that:

View attachment 3669But how can we find the type of the edges?

 

[evinda]

This graph:

View attachment 3678shows the parent of each node.
When there is a red edge does this mean that we have a tree edge? If so, could you explain me why? (Thinking)

Aren't $jh,ia$ forward edges and $ag$ a back edge? (Thinking)