Question $1(15$ points$)$ Recall that a shortest path between two nodes $x$ and $Y$ is any path of minimum length between them. We say that a node $Z$ is pivotal for a pair of distinct nodes $x$ and $Y$ if $Z$ lies on every shortest path between $x$ and $Y,$ and $Z$ is not $x$ or $Y.$

$($a$)$ Construct a network on $6$ nodes where one node, $x,$ is pivotal for every pair of nodes not including $x$ and every other node is pivotal for no pairs.

$($b$)$ Starting with an empty network on $8$ nodes, if we sequentially add one edge at a time and check for pivotal nodes, how many edges can we add without creating a pivotal node?

$($c$)$ If we don't have to add edges one at a time, but instead can add a set of edges all at once, how many edges can we place in a network with $8$ nodes such that no node is pivotal?