Strong geodetic number of complete bipartite graphs, crown graphs and hypercubes

TL;DR

This paper investigates the strong geodetic number for specific graph classes, providing formulas for complete bipartite graphs and crown graphs, and discussing bounds for hypercubes, advancing understanding of vertex coverage in these structures.

Contribution

It derives explicit formulas for the strong geodetic number of complete bipartite and crown graphs, and explores bounds for hypercubes, extending previous results in graph theory.

Findings

Formula for $ ext{sg}(K_{n,m})$ in complete bipartite graphs

Formula for $ ext{sg}(S_n^0)$ in crown graphs

Bounds on $ ext{sg}(Q_n)$ for hypercubes

Abstract

The strong geodetic number, $\text{sg}(G),$ of a graph $G$ is the smallest number of vertices such that by fixing one geodesic between each pair of selected vertices, all vertices of the graph are covered. In this paper, the study of the strong geodetic number of complete bipartite graphs is continued. The formula for $\text{sg}(K_{n,m})$ is given, as well as a formula for the crown graphs $S_n^0$. Bounds on $\text{sg}(Q_n)$ are also discussed.