6F. DFS Good Pairs

The DFS orders are $(1,2,4,3)$ and $(1,3,2,4)$. Vertex $3$ is never the third vertex, so no pair of the form $(3,j)$ is good. Vertex $2$ can be second after vertex $1$. Vertex $4$ can be fourth after all smaller vertices. Therefore the good pairs are $(2,1)$, $(4,1)$, $(4,2)$, and $(4,3)$.

The DFS orders are $(1,2,4,5,3)$, $(1,2,5,4,3)$, $(1,3,2,4,5)$, and $(1,3,2,5,4)$. Vertex $3$ is never the third vertex. Vertex $4$ can be fourth in $(1,3,2,4,5)$, where vertices $1$, $2$, and $3$ are before it. Vertex $5$ can be fifth in $(1,3,2,4,5)$, where all smaller vertices are before it. Together with $(2,1)$, this gives $8$ good pairs.

Vertex $4$ is never the fourth vertex. Vertex $3$ can be third, for example in $(1,2,3,4,5,6)$, but then among smaller vertices only vertex $1$ is before it. Vertex $5$ can be fifth after all smaller vertices, for example in $(1,3,4,2,5,6)$. Vertex $6$ can be sixth after all smaller vertices, for example in $(1,3,4,2,5,6)$. Therefore the answer is $1+1+4+5=11$.

The DFS orders are $(1,2,4,5,3,6)$, $(1,2,5,4,3,6)$, $(1,3,6,2,4,5)$, and $(1,3,6,2,5,4)$. Vertex $3$ is never the third vertex. Vertex $4$ can be fourth only in orders starting with $(1,3,6,2)$, so only vertices $1$ and $2$ may be smaller vertices before it. Vertex $5$ can be fifth with vertices $1$, $2$, and $3$ before it. Vertex $6$ can be sixth after all smaller vertices, for example in $(1,2,4,5,3,6)$. Together with $(2,1)$, this gives $11$ good pairs.

Every DFS order starts by completely visiting either the subtree of vertex $2$ or the subtree of vertex $3$. Therefore vertices $4$ and $5$ are never in their own numbered positions. Vertex $6$ can be sixth after all smaller vertices, for example in $(1,2,4,5,3,6,7)$. Vertex $7$ can be seventh after all smaller vertices, for example in $(1,2,4,5,3,6,7)$. Together with $(2,1)$, this gives $12$ good pairs.