# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2017-2018): c. 84%

# Discussiones Mathematicae Graph Theory

## NOTES ON THE INDEPENDENCE NUMBER IN THE CARTESIAN PRODUCT OF GRAPHS

G. Abay-Asmerom, R. Hammack, C.E. Larson and D.T. Taylor

Department of Mathematics and Applied Mathematics
Virginia Commonwealth University
Richmond, VA 23284, USA

## Abstract

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.

Keywords: independence number, Cartesian product, critical independent set, radius, r-ciliate.

2010 Mathematics Subject Classification: 05C69.

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