Discussiones Mathematicae Graph Theory 31(1) (2011) 25-35
doi: 10.7151/dmgt.1527

[BIBTex] [PDF] [PS]

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.

References

[1] B. Bresar and B. Zmazek, On the Independence Graph of a Graph, Discrete Math. 272 (2003) 263-268, doi: 10.1016/S0012-365X(03)00194-8.
[2] S. Butenko and S. Trukhanov, Using Critical Sets to Solve the Maximum Independent Set Problem, Operations Research Letters 35 (2007) 519-524.
[3] E. DeLaVina, C.E. Larson, R. Pepper and B. Waller, A Characterization of Graphs Where the Independence Number Equals the Radius, submitted, 2009.
[4] P. Erdös, M. Saks and V. Sós, Maximum Induced Trees in Graphs, J. Combin. Theory (B) 41 (1986) 61-69, doi: 10.1016/0095-8956(86)90028-6.
[5] S. Fajtlowicz, A Characterization of Radius-Critical Graphs, J. Graph Theory 12 (1988) 529-532, doi: 10.1002/jgt.3190120409.
[6] S. Fajtlowicz, Written on the Wall: Conjectures of Graffiti, available on the WWW at: http://www.math.uh.edu/~clarson/graffiti.html.
[7] M. Garey and D. Johnson, Computers and Intractability (W.H. Freeman and Company, New York, 1979).
[8] J. Hagauer and S. Klavžar, On Independence Numbers of the Cartesian Product of Graphs, Ars. Combin. 43 (1996) 149-157.
[9] P. Hell, X. Yu and H. Zhou, Independence Ratios of Graph Powers, Discrete Math. 127 (1994) 213-220, doi: 10.1016/0012-365X(92)00480-F.
[10] W. Imrich and S. Klavžar, Product Graphs (Wiley-Interscience, New York, 2000).
[11] W. Imrich, S. Klavžar and D. Rall, Topics in Graph Theory: Graphs and their Cartesian Product, A K Peters (Wellesley, MA, 2008).
[12] P.K. Jha and G. Slutzki, Independence Numbers of Product Graphs, Appl. Math. Lett. 7 (1994) 91-94, doi: 10.1016/0893-9659(94)90018-3.
[13] S. Klavžar, Some New Bounds and Exact Results on the Independence Number of Cartesian Product Graphs, Ars Combin. 74 (2005) 173-186.
[14] C.E. Larson, A Note on Critical Independent Sets, Bulletin of the Institute of Combinatorics and its Applications 51 (2007) 34-46.
[15] C.E. Larson, Graph Theoretic Independence and Critical Independent Sets, Ph.D. Dissertation (University of Houston, 2008).
[16] C.E. Larson, The Critical Independence Number and an Independence Decomposition, submitted, 2009 (available at www.arxiv.org: arXiv:0912.2260v1).
[17] L. Lovász and M.D. Plummer, Matching Theory (North Holland, Amsterdam, 1986).
[18] M.P. Scott, J.S. Powell and D.F. Rall, On the Independence Number of the Cartesian Product of Caterpillars, Ars Combin. 60 (2001) 73-84.
[19] C.-Q. Zhang, Finding Critical Independent Sets and Critical Vertex Subsets are Polynomial Problems, SIAM J. Discrete Math. 3 (1990) 431-438, doi: 10.1137/0403037.

Received 27 July 2009
Revised 10 March 2010
Accepted 11 March 2010