Characterizing Cartesian fixers and multipliers

Stephen 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 -