Geodesic packing in graphs

Abstract

A geodesic packing of a graph G is a set of vertex-disjoint maximal geodesics. The maximum cardinality of a geodesic packing is the geodesic packing number gpack(G). It is proved that the decision version of the geodesic packing number is NP-complete. We also consider the geodesic transversal number, gt(G), which is the minimum cardinality of a set of vertices that hit all maximal geodesics in G. While gt(G)≥gpack(G) in every graph G, the quotient gt(G)/gpack(G) is investigated. By using the rook's graph, it is proved that there does not exist a constant C<3 such that gt(G)gpack(G)≤C would hold for all graphs G. If T is a tree, then it is proved that gpack(T)=gt(T), and a linear algorithm for determining gpack(T) is derived. The geodesic packing number is also determined for the strong product of paths.