The geodesic-transversal problem

TL;DR

This paper introduces the geodesic-transversal problem, proving its NP-completeness and providing efficient algorithms for trees and spread cactus graphs, advancing understanding of vertex sets intersecting all maximal shortest paths.

Contribution

It formally defines the geodesic-transversal problem, proves its NP-completeness, and develops fast algorithms for specific graph classes like trees and spread cactus graphs.

Findings

gt(G)=1 iff G is a subdivided star

The problem is NP-complete in general

Efficient algorithms are provided for trees and spread cactus graphs

Abstract

A maximal geodesic in a graph is a geodesic (alias shortest path) which is not a subpath of a longer geodesic. The geodesic-transversal problem in a graph $G$ is introduced as the task to find a smallest set $S$ of vertices of $G$ such that each maximal geodesic has at least one vertex in $S$. The minimum cardinality of such a set is the geodesic-transversal number ${\rm gt}(G)$ of $G$. It is proved that ${\rm gt}(G) = 1$ if and only if $G$ is a subdivided star and that the geodesic-transversal problem is NP-complete. Fast algorithms to determine the geodesic-transversal number of trees and of spread cactus graphs are designed, respectively.