Edge-Isoperimetric Problems for Cartesian Powers of Regular Graphs

Edge-Isoperimetri Problems for Cartesian Powers of Regular Graphs

The vertex isoperimetric problem for the powers of the diamond graph

We introduce a new graph for all whose cartesian powers the vertex isoperimetric problem has nested solutions. This is the fourth kind of graphs with this property besides the well-studied graphs like hypercubes, grids and tori. In contrast to the mentioned graphs, our graph is not bipartite. We present an exact solution to the vertex isoperimetric problem on our graph by introducing a new clas...

On an equivalence in discrete extremal problems

We introduce some equivalence relations on graphs and posets and prove that they are closed under the cartesian product operation. These relations concern the edge-isoperimetric problem on graphs and the shadow minimization problems on posets. For a long time these problems have been considered quite independently. We present close connections between them. In particular we show that a number o...

A local-global principle for vertex-isoperimetric problems

We consider the vertex-isoperimetric problem for cartesian powers of a graph G. A total order on the vertex set of G is called isoperimetric if the boundary of sets of a given size k is minimum for any initial segment of , and the ball around any initial segment is again an initial segment of . We prove a local-global principle with respect to the so-called simplicial order on Gn (see Section 2...

An Edge-Isoperimetric Problem for Powers of the Petersen Graph

In this paper we introduce a new order on the set of n-dimensional tuples and prove that this order preserves nestedness in the edge isoperimetric problem for the graph Pn, defined as the nth cartesian power of the well-known Petersen graph. The cutwidth and wirelength of Pn are also derived. These results are then generalized for the cartesian product of Pn and the m-dimensional binary hypercube.