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## Discussiones Mathematicae Graph Theory

2012 | 32 | 1 | 161-175
Tytuł artykułu

### Characterizing Cartesian fixers and multipliers

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Let G ☐ H denote the Cartesian product of the graphs G and H. In 2004, Hartnell and Rall [On dominating the Cartesian product of a graph and K₂, Discuss. Math. Graph Theory 24(3) (2004), 389-402] characterized prism fixers, i.e., graphs G for which γ(G ☐ K₂) = γ(G), and noted that γ(G ☐ Kₙ) ≥ min{|V(G)|, γ(G)+n-2}. We call a graph G a consistent fixer if γ(G ☐ Kₙ) = γ(G)+n-2 for each n such that 2 ≤ n < |V(G)|- γ(G)+2, and characterize this class of graphs.
Also in 2004, Burger, Mynhardt and Weakley [On the domination number of prisms of graphs, Dicuss. Math. Graph Theory 24(2) (2004), 303-318] characterized prism doublers, i.e., graphs G for which γ(G ☐ K₂) = 2γ(G). In general γ(G ☐ Kₙ) ≤ nγ(G) for any n ≥ 2. We call a graph attaining equality in this bound a Cartesian n-multiplier and also characterize this class of graphs.
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Kategorie tematyczne
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Czasopismo
Rocznik
Tom
Numer
Strony
161-175
Opis fizyczny
Daty
wydano
2012
otrzymano
2009-02-26
poprawiono
2011-03-15
zaakceptowano
2011-04-04
Twórcy
autor
• Department of Mathematics and Statistics, University of Victoria, P.O. Box 3060 STN CSC, Victoria, B.C., Canada V8W 3R4
• Department of Mathematics and Statistics, University of Victoria, P.O. Box 3060 STN CSC, Victoria, B.C., Canada V8W 3R4
Bibliografia
• [1] A.P. Burger, C.M. Mynhardt and W.D. Weakley, On the domination number of prisms of graphs, Dicuss. Math. Graph Theory 24 (2004) 303-318, doi: 10.7151/dmgt.1233.
• [2] G. Chartrand and F. Harary, Planar permutation graphs, Ann. Inst. H. Poincaré Sect. B (N.S.) 3 (1967) 433-438.
• [3] B.L. Hartnell and D.F. Rall, Lower bounds for dominating Cartesian products, J. Combin. Math. Combin. Comput. 31 (1999) 219-226.
• [4] B.L. Hartnell and D.F. Rall, On dominating the Cartesian product of a graph and K₂, Discuss. Math. Graph Theory 24 (2004) 389-402, doi: 10.7151/dmgt.1238.
• [5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).
• [6] C.M. Mynhardt and Z. Xu, Domination in prisms of graphs: Universal fixers, Utilitas Math. 78 (2009) 185-201.
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Bibliografia
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