Geodesic packing in graphs

TL;DR

This paper studies the properties and computational complexity of geodesic packings and transversals in graphs, proving NP-completeness, exploring ratios, and providing algorithms for trees and product graphs.

Contribution

It introduces the concept of geodesic packing and transversal numbers, proves NP-completeness, and provides exact and algorithmic results for specific graph classes.

Findings

Decision problem for geodesic packing is NP-complete.

For trees, geodesic packing number equals the transversal number, with a linear-time algorithm.

The ratio of transversal to packing numbers can be arbitrarily close to 3.

Abstract

Given a graph $G$, a geodesic packing in $G$ is a set of vertex-disjoint maximal geodesics, and the geodesic packing number of $G$, ${\gpack}(G)$, 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, ${\gt}(G)$, which is the minimum cardinality of a set of vertices that hit all maximal geodesics in $G$. While $\gt(G)\ge \gpack(G)$ in every graph $G$, the quotient ${\rm gt}(G)/{\rm gpack}(G)$ is investigated. By using the rook's graph, it is proved that there does not exist a constant $C < 3$ such that $\frac{{\rm gt}(G)}{{\rm gpack}(G)}\le C$ would hold for all graphs $G$. If $T$ is a tree, then it is proved that ${\rm gpack}(T) = {\rm gt}(T)$, and a linear algorithm for determining ${\rm gpack}(T)$ is derived. The geodesic packing number is also determined for the strong product of paths.