Convex universal fixers

• Autorzy: Magdalena Lemańska , Rita Zuazua
• Języki publikacji: EN

Abstrakty

EN

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 $E(πG) = E(G) ∪ E(G') ∪ M_{π}$, 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.

Słowa kluczowe

EN

convex sets; dominating sets; universal fixers

Kategorie tematyczne

• 05C69: Dominating sets, independent sets, cliques
• 05C99: None of the above, but in this section

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2012
• Tom: 32
• Numer: 4
• Strony: 807-812

Daty

wydano

2012

otrzymano

2011-08-25

poprawiono

2011-11-14

zaakceptowano

2011-11-18

Twórcy

autor

Magdalena Lemańska

• Gdańsk University of Technology, Narutowicza 11/12, 80-233 Gdańsk, Poland

autor

Rita Zuazua

• Departamento de Matematicas, Facultad de Ciencias, UNAM, Mexico

Bibliografia

• [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.
• [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.
• [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.
• [4] R.G. Gibson, Bipartite graphs are not universal fixers, Discrete Math. 308 (2008) 5937-5943, doi: 10.1016/j.disc.2007.11.006.
• [5] M. Lemańska, Weakly convex and convex domination numbers, Opuscula Math. 24 (2004) 181-188.
• [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.
• [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.
• [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.
• [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.

Identyfikatory

DOI

10.7151/dmgt.1631