Algorithmic upper bounds for graph geodetic number

TL;DR

This paper introduces new algorithmic methods to compute upper bounds for the geodetic number in graphs, a measure related to shortest path coverage, demonstrating efficiency across various graph structures.

Contribution

It proposes novel algorithms inspired by integer linear programming to estimate upper bounds for the graph geodetic number.

Findings

Algorithms efficiently compute upper bounds on the geodetic number.

Methods perform well on structurally diverse graphs.

Demonstrated effectiveness through computational experiments.

Abstract

Graph theoretical problems based on shortest paths are at the core of research due to their theoretical importance and applicability. This paper deals with the geodetic number which is a global measure for simple connected graphs and it belongs to the path covering problems: what is the minimal-cardinality set of vertices, such that all shortest paths between its elements cover every vertex of the graph. Inspired by the exact 0-1 integer linear programming formalism from the recent literature, we propose a new methods to obtain upper bounds for the geodetic number in an algorithmic way. The efficiency of these algorithms are demonstrated on a collection of structurally different graphs.