Convex universal fixers

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: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 K n .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.

Convex and weakly convex domination in prism graphs

“…Since so many of the results in [3] can be generalized to weakly convex domination, one could expect a weakly convex analogue of Theorem 4 to be true as well. However, Lemańska and Zuazua's proof relies on showing that if G is a conncected graph with diam(G) ≤ 2, then a dominating set of πG with less than |V G | vertices cannot be convex.…”

“…Convex domination in prism graphs was studied by Lemańska and Zuazua in [3], where they prove the following theorem.…”

The convex and weak convex domination number of convex polytopes

“…The convex domination number was defined during verbal communication between Jerzy Topp and Magdalena Lemanska in 2002 (stated in [15]). In [18] it was proven that decision problems of WCDSP and CDSP are NP-complete even for bipartite and split graphs.…”