The Well-Covered Dimension of Products of Graphs

TL;DR

This paper investigates the well-covered dimension of various graph products, providing formulas, bounds, and methods for calculating this property in complex graph constructions.

Contribution

It introduces new formulas and bounds for the well-covered dimension of Cartesian products, vertex blowups, and lexicographic products of graphs.

Findings

Derived formulas for well-covered dimension of Cartesian products.

Established bounds for the dimension of $K_n \times G$.

Provided methods for vertex blowups and lexicographic products.

Abstract

We discuss how to find the well-covered dimension of a graph that is the Cartesian product of paths, cycles, complete graphs, and other simple graphs. Also, a bound for the well-covered dimension of $K_n\times G$ is found, provided that $G$ has a largest greedy independent decomposition of length $c<n$. Formulae to find the well-covered dimension of graphs obtained by vertex blowups on a known graph, and to the lexicographic product of two known graphs are also given.