# Convex universal fixers

Magdalena Lemańska; Rita Zuazua

Discussiones Mathematicae Graph Theory (2012)

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

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topMagdalena 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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- [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] 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] 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] M. Lemańska, Weakly convex and convex domination numbers, Opuscula Math. 24 (2004) 181-188. Zbl1076.05060
- [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] 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] 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] C.M. Mynhardt and Z. Xu, Domination in Prisms of Graphs: Universal Fixers, Util. Math. 78 (2009) 185-201. Zbl1284.05199
- [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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