Graph products and integer domination

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We consider the κkκ-domination number γκ kκ(G) of a graph G and the Cartesian product G□H and the strong direct product Gsquared timesH of graphs G and H. We prove that for integers k,m<1, γκ kκ(Gsquared timesH)<γκ γκkκ(H) κ(G) and γκ kmκ(Gsquared timesH)≤γκ kκ(G)γκ mκ(H), from which earlier results obtained by Fisher on γ(Gsquared timesH) and Fisher et al. on the fractional domination number γf(Gsquared timesH) were derived. We extend a result from Brešar et al. on γ(G□H) for claw-free graphs G. We also point out some sufficient conditions for graphs to satisfy the generalized form of Vizing's conjecture suggested by Hou and Lu. © 2012 Elsevier B.V. All rights reserved.




John, N., & Suen, S. (2013). Graph products and integer domination. Discrete Mathematics, 313(3), 217–224.

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