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 dont 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?