# Convex universal fixers

• Volume: 32, Issue: 4, page 807-812
• ISSN: 2083-5892

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## Abstract

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In [1] Burger and Mynhardt introduced the idea of universal fixers. Let G = (V, E) be a graph with n vertices and G’ a copy of G. For a bijective function π: V(G) → V(G’), define the prism πG of G as follows: V(πG) = V(G) ∪ V(G’) and $E\left(\pi G\right)=E\left(G\right)\cup E\left({G}^{\text{'}}\right)\cup {M}_{\pi }$, where ${M}_{\pi }=u\pi \left(u\right)|u\in V\left(G\right)$. Let γ(G) be the domination number of G. If γ(πG) = γ(G) for any bijective function π, then G is called a universal fixer. In [9] it is conjectured that the only universal fixers are the edgeless graphs K̅ₙ. In this work we generalize the concept of universal fixers to the convex universal fixers. In the second section we give a characterization for convex universal fixers (Theorem 6) and finally, we give an in infinite family of convex universal fixers for an arbitrary natural number n ≥ 10.

## How to cite

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Magdalena Lemańska, and Rita Zuazua. "Convex universal fixers." Discussiones Mathematicae Graph Theory 32.4 (2012): 807-812. <http://eudml.org/doc/271021>.

@article{MagdalenaLemańska2012,
abstract = {In [1] Burger and Mynhardt introduced the idea of universal fixers. Let G = (V, E) be a graph with n vertices and G’ a copy of G. For a bijective function π: V(G) → V(G’), define the prism πG of G as follows: V(πG) = V(G) ∪ V(G’) and $E(πG) = E(G) ∪ E(G^\{\prime \}) ∪ M_\{π\}$, where $M_\{π\} = \{u π(u) | u ∈ V(G)\}$. Let γ(G) be the domination number of G. If γ(πG) = γ(G) for any bijective function π, then G is called a universal fixer. In [9] it is conjectured that the only universal fixers are the edgeless graphs K̅ₙ. In this work we generalize the concept of universal fixers to the convex universal fixers. In the second section we give a characterization for convex universal fixers (Theorem 6) and finally, we give an in infinite family of convex universal fixers for an arbitrary natural number n ≥ 10.},
author = {Magdalena Lemańska, Rita Zuazua},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {convex sets; dominating sets; universal fixers},
language = {eng},
number = {4},
pages = {807-812},
title = {Convex universal fixers},
url = {http://eudml.org/doc/271021},
volume = {32},
year = {2012},
}

TY - JOUR
AU - Magdalena Lemańska
AU - Rita Zuazua
TI - Convex universal fixers
JO - Discussiones Mathematicae Graph Theory
PY - 2012
VL - 32
IS - 4
SP - 807
EP - 812
AB - In [1] Burger and Mynhardt introduced the idea of universal fixers. Let G = (V, E) be a graph with n vertices and G’ a copy of G. For a bijective function π: V(G) → V(G’), define the prism πG of G as follows: V(πG) = V(G) ∪ V(G’) and $E(πG) = E(G) ∪ E(G^{\prime }) ∪ M_{π}$, where $M_{π} = {u π(u) | u ∈ V(G)}$. Let γ(G) be the domination number of G. If γ(πG) = γ(G) for any bijective function π, then G is called a universal fixer. In [9] it is conjectured that the only universal fixers are the edgeless graphs K̅ₙ. In this work we generalize the concept of universal fixers to the convex universal fixers. In the second section we give a characterization for convex universal fixers (Theorem 6) and finally, we give an in infinite family of convex universal fixers for an arbitrary natural number n ≥ 10.
LA - eng
KW - convex sets; dominating sets; universal fixers
UR - http://eudml.org/doc/271021
ER -

## References

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1. [1] A.P. Burger, C.M. Mynhardt and W.D. Weakley, On the domination number of prisms of graphs, Discuss. Math. Graph Theory 24 (2004) 303-318, doi: 10.7151/dmgt.1233. Zbl1064.05111
2. [2] A.P. Burger and C.M. Mynhardt, Regular graphs are not universal fixers, Discrete Math. 310 (2010) 364-368, doi: 10.1016/j.disc.2008.09.016. Zbl1216.05098
3. [3] E.J. Cockayne, R.G. Gibson and C.M. Mynhardt, Claw-free graphs are not universal fixers, Discrete Math. 309 (2009) 128-133, doi: 10.1016/j.disc.2007.12.053. Zbl1219.05116
4. [4] R.G. Gibson, Bipartite graphs are not universal fixers, Discrete Math. 308 (2008) 5937-5943, doi: 10.1016/j.disc.2007.11.006. Zbl1181.05068
5. [5] M. Lemańska, Weakly convex and convex domination numbers, Opuscula Math. 24 (2004) 181-188. Zbl1076.05060
6. [6] J. Cyman, M. Lemańska and J. Raczek, Graphs with convex domination number close to their order, Discuss. Math. Graph Theory 26 (2006) 307-316, doi: 10.7151/dmgt.1322. Zbl1140.05302
7. [7] J. Raczek and M. Lemańska, A note of the weakly convex and convex domination numbers of a torus, Discrete Appl. Math. 158 (2010) 1708-1713, doi: 10.1016/j.dam.2010.06.001. Zbl1208.05102
8. [8] M. Lemańska, I. González Yero and J.A. Rodríguez-Velázquez, Nordhaus-Gaddum results for a convex domination number of a graph, Acta Math. Hungar., to appear (2011).
9. [9] C.M. Mynhardt and Z. Xu, Domination in Prisms of Graphs: Universal Fixers, Util. Math. 78 (2009) 185-201. Zbl1284.05199
10. [10] C.M. Mynhardt and M. Schurch, Paired domination in prisms of graphs, Discuss. Math. Graph Theory 31 (2011) 5-23, doi: 10.7151/dmgt.1526. Zbl1238.05201

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