A problem and solution that arise from a graph-theoretic game called Coppers and Robbers are discussed in this well-written paper. An isometric-path number is related to the number of cops that are needed in the graph to catch a thief.

An isometric path between two vertices in a graph is defined as the shortest path joining them. The isometric-path number is defined as the minimum number of isometric paths required to cover all the vertices of the given graph. A block graph is a graph in which each of the blocks (two connected components) is a complete graph. The authors identify the exact values for the isometric-path numbers of block graphs as the smallest integer greater than or equal to half the number of noncut vertices in the graph. They also provide a linear-time algorithm for finding those paths.