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We consider the problem of embedding the n-dimensional cube into a rectangular grid with 2 n vertices in such a way as to minimize the congestion, the maximum number of edges along any point of the grid. After presenting a short solution for the cutwidth problem of the n-cube (in which the n-cube is embedded into a path), we show how to extend the results… (More)
We consider one-to-one embeddings of the n-dimensional hypercube into grids with 2 n vertices and present lower and upper bounds and asymptotic estimates for minimal dilation, edge-congestion, and their mean values. We also introduce and study two new cost-measures for such embeddings, namely the sum over i = 1; :::; n of dilations and the sum of edge… (More)
This paper is a survey on discrete isoperimetric type problems. We present here as some known facts about their solutions as well some new results and demonstrate a general techniques used in this area. The main attention is paid to the unit cube and cube like structures. Besides some applications of the isoperimetric approach are listed too.
We consider the collection of all spanning trees of a graph with distance between them based on the size of the symmetric difference of their edge sets. A central spanning tree of a graph is one for which the maximal distance to all other spanning trees is minimal. We prove that the problem of constructing a central spanning tree is algorithmically… (More)
We survey results on edge isoperimetric problems on graphs, present some new results and show some applications of such problems in combinatorics and computer science.
We consider subsets of the n-dimensional grid with the Manhattan metrics, (i.e. the Cartesian product of chains of lengths k 1 ; : : : ; k n) 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 rst problem was considered for k 1 = = k n by… (More)
We present embeddings of generalized ladders as subgraphs into the hypercube. By embedding caterpillars into ladders, we obtain embeddings of caterpillars into the hypercube. In this way we obtain almost all known results concerning the embed-dings of caterpillars into the hypercube. In addition we construct embeddings for some new types of caterpillars.
We consider the complete t-ary trees with k levels (T k;t) and their embedding as subgraphs into the binary hypercube of possibly small dimension n. This n is called the dimension of T k;t , and it is known if one of k; t equals 2. Here we study the next open case when one of k; t equals 3. We present an asymptotic for the dimension of T 3;t in the form n… (More)
The communication overhead is a major bottleneck for the execution of a process graph on a parallel computer system. In the case of two processors, the minimization of the communication can be modeled by the graph bisection problem. The spectral lower bound of ¾ Î for the bisection width of a graph is well-known. The bisection width is equal to ¾ Î iff all… (More)
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.… (More)