On the geodetic and the hull numbers in strong product graphs

TL;DR

This paper studies the properties of geodetic and hull numbers in strong product graphs, establishing bounds and exact values for these parameters, advancing understanding of graph convexity in complex graph products.

Contribution

It introduces bounds and exact formulas for geodetic and hull numbers in strong product graphs, expanding theoretical knowledge in graph convexity.

Findings

Bounds for geodetic and hull numbers established

Exact values computed for specific strong product graphs

Enhanced understanding of convexity in graph products

Abstract

A set S of vertices of a connected graph G is convex, if for any pair of vertices u; v 2 S, every shortest path joining u and v is contained in S . The convex hull CH(S) of a set of vertices S is defined as the smallest convex set in G containing S. The set S is geodetic, if every vertex of G lies on some shortest path joining two vertices in S, and it is said to be a hull set if its convex hull is V(G). The geodetic and the hull numbers of G are the cardinality of a minimum geodetic and a minimum hull set, respectively. In this work, we investigate the behavior of both geodetic and hull sets with respect to the strong product operation for graphs. We also stablish some bounds for the geodetic number and the hull number and obtain the exact value of these parameters for a number of strong product graphs.