pictorial representation of pre and post ordering for edge types
&quot;I am assuming that $u$ is the node/vertex we see first and then we see $v$ later.&quot; That is a wrong assumption, which makes that summarizing table unintelligible.
The right assumption consists of two parts.

We are given a directed graph that contains $(u,v)$, the edge from $u$ to $v$.
We have just completed one depth first search (DFS) on that graph, which has discovered both $u$ and $v$.

Given the assumption above, that table lists the correspondence between the &quot;pre/post ordering for $u$ and $v$&quot;, and the &quot;edge type&quot; of $(u,v)$. Please note that I rephrase &quot;pre/post ordering for $(u,v)$&quot; as in the table to &quot;pre/post ordering for $u$ and $v$&quot;, since the meaning of &quot;$(u,v)$&quot;, the edge from $u$ to $v$ does not fit the former phrase exactly.

The first row, &quot;$\stackrel{[}{u}\,\stackrel{[}{v}\,\stackrel{]}{v}\,\stackrel{]}{u}\ $ Tree/forward&quot; says that $u$'s discovery time and departure time enclose $v$'s discovery time and departure iff $(u,v)$ is a tree edge or a forward edge. $u$ is visited first in this case.
The second row, &quot;$\stackrel{[}{v}\,\stackrel{[}{u}\,\stackrel{]}{u}\,\stackrel{]}{v}\ $ Back&quot; says that $v$'s discovery time and departure time enclose $u$'s discovery time and departure time iff $(u,v)$ is a back edge. $v$ is visited first in this case.
The third row, &quot;$\stackrel{[}{v}\,\stackrel{]}{v}\,\stackrel{[}{u}\,\stackrel{]}{u}\ $ Cross&quot; says that $v$'s discovery time and departure time are before $u$'s discovery time and departure time iff $(u,v)$ is a cross edge. $v$ is visited first in this case.

Neither do we assume $u$ is visited first nor do we assume $v$ is visited first. The node that is visited first on each row of that table is, of course, the node that appears first from the left on that row.

Exercise (as given in the book). With the same assumption, &quot;do you see why no other orderings are possible?&quot; For example, why is &quot;$\stackrel{[}{v}\,\stackrel{[}{u}\,\stackrel{]}{v}\,\stackrel{]}{u}$&quot; not possible? why is &quot;$\stackrel{[}{u}\,\stackrel{]}{u}\,\stackrel{[}{v}\,\stackrel{]}{v}$&quot; not possible?