[Math] Gatekeeper Graph

Question

From the book Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Definitions are on page 45, questions are on page 46.

A node X is a gatekeeper if, for some other two nodes Y and Z, every path from Y to Z passes through X. (It lies on every path between other pairs of nodes)

A node X is a local gatekeeper if there are two neighbors of X, say Y and Z, that are not connected by an edge. (That is, for X to be a local gatekeeper, there should be two nodes Y and Z such that Y and Z each have edges to X, but not to each other.)

If then,

1. Give an example of a graph in which more than half of all nodes are gatekeepers.

2. Give an example of a graph in which there are no gatekeepers but in which every node is a local gatekeeper

Question 2- Not gatekeeper but every node is a Local Gatekeeper. Whether it is correct?

Question 1- More than half of all nodes are gatekeepers.

Answer 1

One answer to part 1) any line graph on at least 5 vertices and part 2) any cycle graph