On the well-coveredness of Cartesian products of graphs

A graph G is well-covered if every maximal independent set has the same cardinality. This paper investigates when the Cartesian product of two graphs is well-covered. We prove that if G and H both belong to a large class of graphs that includes all non-well-covered triangle-free graphs and most well-covered triangle-free graphs, then G × H is not well-covered. We also show that if G is not well-covered, then neither is G × G . Finally, we show that G × G is not well-covered for all graphs of girth at least 5 by introducing super well-covered graphs and classifying all such graphs of girth at least 5.