Bounding the search number of graph products

TL;DR

This paper investigates the search number of Cartesian graph products, especially focusing on paths and cliques, providing bounds and exact values for these products and their implications for arbitrary graphs.

Contribution

It determines the pathwidth of clique products and establishes bounds for the search number of graph products based on clique numbers, advancing understanding of graph search parameters.

Findings

Pathwidth of clique products is characterized.

Lower bounds for search number of clique products are established.

Bounds for arbitrary graph products based on clique numbers are derived.

Abstract

In this paper, we provide results for the search number of the Cartesian product of graphs. We consider graphs on opposing ends of the spectrum: paths and cliques. Our main result determines the pathwidth of the product of cliques and provides a lower bound for the search number of the product of cliques. A consequence of this result is a bound for the search number of arbitrary graphs G and H based on their respective clique numbers.