$G$ :

Figure $1\mathrm{.}20$: The graph $G$ in Exercises $1\mathrm{.}11$ and $1\mathrm{.}12$

$1\mathrm{.}12$ For the graph $G$ of Figure $1\mathrm{.}20,$ give an example of each of the following

or explain why no such example exists.

$($a$)$ An $U-V$ walk of length $6.$

$($b$)$ A $v-w$ path that is not a $v-w$ path.

$($c$)$ An $r-z$ track of length $2.$

$($d$)$ An $x-z$ track of length $3.$

$($e$)$ An $x-t$ track of length $d(x,t).$

$($f$)$ A circuit of length $10.$

$($g$)$ A cycle of length $8.$

$($h$)$ A geodesic whose length is diam$(G).$