Problem $1.$ Consider the following undirected graph $G_1.$

a$.$ Perform Prim's algorithm on $G_1,$ using $s$ as the startng vertex, to find a minimum spanning tree $($MST$).$ Show the updated key and $$ values for each vertex after each iteration of the while loop. Draw the MST rooted at $s.$

b$.$ Perform Dijkstra's algorithm to find a shortest path from $s$ to every vertex in $G_1.$ Show the updated $d$ and $$ values for each vertex after each iteration of the while loop. Draw the shortest$-$paths tree rooted at $s.$

c$.$ Is the shortest$-$path tree you obtained in part b also an MST of $G_1?$ Breifly explain why or why not.