$10$ points

Problem $4.$

Find an MST using Prim's algorithm, starting from the vertex a$,$ for the following graph:

$(3$ points$)$ Darken the edges of the resulting MST$.$

$(2$ points$)$ List the order in which the vertices were extracted from the queue Q $.$

$(5$ points$)$ Show a table with values for $\backslash (\backslash$mathrm$\{$v$\}.\backslash$mathrm$\{$key$\}\backslash )$ and $\backslash (\backslash$mathrm$\{$v$\}.\backslash$pi $\backslash )$ for each $\backslash (\backslash$mathrm$\{$v$\}\backslash$in $\backslash$mathrm$\{$V$\}\backslash ),$ at each iterative step.