# Characterizing Cartesian fixers and multipliers

Stephen Benecke; Christina M. Mynhardt

Discussiones Mathematicae Graph Theory (2012)

- Volume: 32, Issue: 1, page 161-175
- ISSN: 2083-5892

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topStephen Benecke, and Christina M. Mynhardt. "Characterizing Cartesian fixers and multipliers." Discussiones Mathematicae Graph Theory 32.1 (2012): 161-175. <http://eudml.org/doc/270897>.

@article{StephenBenecke2012,

abstract = {
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.
},

author = {Stephen Benecke, Christina M. Mynhardt},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {Cartesian product; prism fixer; Cartesian fixer; prism doubler; Cartesian multiplier; domination number; consistent fixer},

language = {eng},

number = {1},

pages = {161-175},

title = {Characterizing Cartesian fixers and multipliers},

url = {http://eudml.org/doc/270897},

volume = {32},

year = {2012},

}

TY - JOUR

AU - Stephen Benecke

AU - Christina M. Mynhardt

TI - Characterizing Cartesian fixers and multipliers

JO - Discussiones Mathematicae Graph Theory

PY - 2012

VL - 32

IS - 1

SP - 161

EP - 175

AB -
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.

LA - eng

KW - Cartesian product; prism fixer; Cartesian fixer; prism doubler; Cartesian multiplier; domination number; consistent fixer

UR - http://eudml.org/doc/270897

ER -

## References

top- [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. Zbl1064.05111
- [2] G. Chartrand and F. Harary, Planar permutation graphs, Ann. Inst. H. Poincaré Sect. B (N.S.) 3 (1967) 433-438. Zbl0162.27605
- [3] B.L. Hartnell and D.F. Rall, Lower bounds for dominating Cartesian products, J. Combin. Math. Combin. Comput. 31 (1999) 219-226. Zbl0938.05048
- [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. Zbl1063.05107
- [5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998). Zbl0890.05002
- [6] C.M. Mynhardt and Z. Xu, Domination in prisms of graphs: Universal fixers, Utilitas Math. 78 (2009) 185-201. Zbl1284.05199

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