Every connected graph G with radius r(G) and independence number α(G) obeys α(G) ≥ r(G). Recently the graphs for which equality holds have been classified. Here we investigate the members of this class that are Cartesian products. We show that for non-trivial graphs G and H, α(G ☐ H) = r(G ☐ H) if and only if one factor is a complete graph on two vertices, and the other is a nontrivial complete graph. We also prove a new (polynomial computable) lower bound α(G ☐ H) ≥ 2r(G)r(H) for the independence number and we classify graphs for which equality holds. The second part of the paper concerns independence irreducibility. It is known that every graph G decomposes into a König-Egervary subgraph (where the independence number and the matching number sum to the number of vertices) and an independence irreducible subgraph (where every non-empty independent set I has more than |I| neighbors). We examine how this decomposition relates to the Cartesian product. In particular, we show that if one of G or H is independence irreducible, then G ☐ H is independence irreducible.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1527,
author = {G. Abay-Asmerom and R. Hammack and C.E. Larson and D.T. Taylor},
title = {Notes on the independence number in the Cartesian product of graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {31},
year = {2011},
pages = {25-35},
zbl = {1284.05190},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1527}
}
G. Abay-Asmerom; R. Hammack; C.E. Larson; D.T. Taylor. Notes on the independence number in the Cartesian product of graphs. Discussiones Mathematicae Graph Theory, Tome 31 (2011) pp. 25-35. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1527/
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