# On well-covered Cartesian products

## TL;DR

This paper investigates the conditions under which Cartesian products of graphs are well-covered, providing classifications for certain classes of graphs based on girth and connectivity.

## Contribution

It classifies all well-covered Cartesian products of nontrivial, connected graphs with girth at least 4 or 5, identifying specific graph pairs that satisfy these conditions.

## Key findings

- If the Cartesian product of two nontrivial, connected graphs of girth ≥ 4 is well-covered, then one graph is K2.
- K2 × K2 and C5 × K2 are the only well-covered Cartesian products with girth ≥ 5.
- The paper extends understanding of well-covered graphs in Cartesian products.

## Abstract

In 1970, Plummer defined a well-covered graph to be a graph $G$ in which all maximal independent sets are in fact maximum. Later Hartnell and Rall showed that if the Cartesian product $G \Box H$ is well-covered, then at least one of $G$ or $H$ is well-covered. In this paper, we consider the problem of classifying all well-covered Cartesian products. In particular, we show that if the Cartesian product of two nontrivial, connected graphs of girth at least $4$ is well-covered, then at least one of the graphs is $K_2$. Moreover, we show that $K_2 \Box K_2$ and $C_5 \Box K_2$ are the only well-covered Cartesian products of nontrivial, connected graphs of girth at least $5$.