Geodesic packing in graphs

10/27/2022

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by   Paul Manuel, et al.

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Given a graph G, a geodesic packing in G is a set of vertex-disjoint maximal geodesics, and the geodesic packing number of G, (G), is the maximum cardinality of a geodesic packing in G. It is proved that the decision version of the geodesic packing number is NP-complete. We also consider the geodesic transversal number, (G), which is the minimum cardinality of a set of vertices that hit all maximal geodesics in G. While (G)≥(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.