Convex universal fixers

Lemańska, Magdalena ; Zuazua, Rita

Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 4, pp. 807-812

Résumé

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.

Détail

Keywords: convex sets, dominating sets, universal fixers