On the Cartesian product of non well-covered graphs

TL;DR

This paper proves that if the Cartesian product of two graphs is well-covered, then at least one of the graphs must be well-covered, answering an open question in graph theory.

Contribution

It establishes a necessary condition for the Cartesian product of graphs to be well-covered, advancing understanding of graph product properties.

Findings

If the Cartesian product of two graphs is well-covered, then at least one factor graph is well-covered.

The paper answers an open question posed by Topp and Volkmann.

Provides new insights into the structure of well-covered graph products.

Abstract

A graph is well-covered if every maximal independent set has the same cardinality, namely the vertex independence number. We answer a question of Topp and Volkmann and prove that if the Cartesian product of two graphs is well-covered, then at least one of them must be well-covered.