$1.$ Find the path with the shortest distance between nodes $1$ and $13$ for the following network, where dig denotes the distance $($in hours$)$ between nodes i and j $(30)$:

d$1,2=5,$ d$1,3=6,$ d$1,4=3,$ d$1,6=12$

d$2,5=7,$ d$2,6=11,2,7=13$

d$3\mathrm{.}5=4,$ d$3\mathrm{.}6=8,$ d$3\mathrm{.}7=5$

d$1,5=4,4,6=3,4,13=15$

ds$,7=8,$ ds$,8=7,$ d$5,9=7$

d$6\mathrm{.}7=5,$ d$6,8=4,$ d$6,9=6$

d$7,10=4,$ d$7\mathrm{.}11=9,$ d$7,12=5$

ds$,10=5,$ ds$\mathrm{.}11=8,$ d$8\mathrm{.}12=7$

d$9,10=3,$ d$9\mathrm{.}12=1$

d$10,13=3$

d$11,13=2$

d$/2,13=3.$

$2.$ a$.$ Refer to problem $1.$ Find the path with the shortest distance between nodes $2$ and $13.(10)$

b$.$ Refer to problem $1.$ If the link between nodes $4$ and $5$ is eliminated $($other parameters unchanged$),$ find the path with the shortest distance between nodes $1$ and $13.$