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arXiv · 1703.08716

On well-covered Cartesian products

Bert L. Hartnell·Douglas F. Rall·Kirsti Wash
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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 $5$.

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Bert L. Hartnell, Douglas F. Rall, Kirsti Wash. 2017-03-25. On well-covered Cartesian products. https://arxiv.org/abs/1703.08716

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