We consider subsets of the n-dimensional grid with the Manhattan metrics, (i.e., the Cartesian product of chains of lengths kl,..., kn) and study those of them which have maximal number of induced edges of the grid, and those which are separable from their complement by the least number of edges. The first problem was considered for kl ..... kn by Bollob~is and Leader [1].Here we extend their result to arbitrary kl,... ,kn, and give also a simpler proof based on a new approach. For the second problem, [1] offers only an inequality. We show that our approach to the first problem also gives a solution for the second problem, if all ki = co. If all ki's axe finite, we present an exact solution for n = 2.
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 P n , defined as the nth cartesian power of the well-known Petersen graph. The cutwidth and wirelength of P n are also derived. These results are then generalized for the cartesian product of P n and the m-dimensional binary hypercube. Problem 1.2. Find a subset of vertices of a given graph, such that the number of edges in the subgraph induced by this subset is maximal among all induced subgraphs with the same number of vertices.
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