$4.$ Suppose we want to find the minimum spanning tree of the following graph.

$($a$)$ Run Kruskal's algorithm $($in case of ties add the edge which is lexicographically first, that is$,$ comes first in the table$).$ Show how the disjoint$-$sets data structure looks at every intermediate stage $($including the structure of the directed trees$),$ assuming path compression is not used.

$($b$)$ Run Prim's algorithm; whenever there is a choice of nodes, always use alphabetic ordering $($e$.$g$.,$ start from node A $).$ Draw a table showing the intermediate values of the cost$/$prey values.