Strong geodetic number of complete bipartite graphs and of graphs with specified diameter

TL;DR

This paper investigates the strong geodetic number, a variant of the classical geodetic problem, focusing on properties related to graph diameter and providing solutions for balanced complete bipartite graphs.

Contribution

It introduces new properties of the strong geodetic problem and explicitly solves it for balanced complete bipartite graphs.

Findings

Characterized strong geodetic number in relation to graph diameter

Solved the problem for balanced complete bipartite graphs

Established general properties of the strong geodetic problem

Abstract

The strong geodetic problem is a recent variation of the classical geodetic problem. For a graph $G$, its strong geodetic number ${\rm sg}(G)$ is the cardinality of a smallest vertex subset $S$, such that each vertex of $G$ lies on one fixed geodesic between a pair of vertices from $S$. In this paper, some general properties of the strong geodesic problem are studied, especially in connection with diameter of a graph. The problem is also solved for balanced complete bipartite graphs.