The Isometric Path Number of a Graph

An isometric path is merely any shortest path between two vertices. Inspired by the game of `Cops and Robber’ and a result by Aigner \(\&\) Fromme [1], we are interested in determining the minimum number of isometric paths required to cover the vertices of a graph. We find a lower bound on this number in terms of the diameter of a graph and find the exact number for trees and grid graphs.