One reason for graph theory’s power as a modeling tool is the fluidity with which one can use it to formalize the properties of large systems using the language of graphs, and then to systematically explore their consequences. In this first set of questions, we work through an example of this process using the concept of a pivotal node.

Recall from earlier in the chapter that a shortest path between two nodes is a path of the minimum possible length. We say that a node X is pivotal for a pair of distinct nodes Y and Z if X lies on every shortest path between Y and Z (and X is not equal to either Y or Z).

For example, in the graph in Figure 2.13, node B is pivotal for two pairs: the pair consisting of A and C, and the pair consisting of A and D. [Notice that B is not pivotal for the pair consisting of D and E because two different shortest paths connect D and E, one of which (using C and F) does not pass through B. Therefore, B is not on every shortest path between D and E.] As another example, note that node D is not pivotal for any pairs.

(a) Give an example of a graph in which every node is pivotal for at least one pair of nodes. Explain your answer.

(b) Give an example of a graph in which every node is pivotal for at least two different pairs of nodes. Explain your answer.

(c) Give an example of a graph having at least four nodes in which there exists a single node X that is pivotal for every pair of nodes (not counting pairs that include X). Explain your answer.