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Z. S. Lozhkina

On optimal exact coverings of a graph in the class of weakly dense bases

We consider the problem on covering a non-oriented connected graph without loops and multiple edges by graphs of arbitrary finite bases. We introduce the notions of complexity of a covering, complexity of a graph and the Shannon function. In accordance with the asymptotic behaviour of the Shannon function, we introduce two classes of bases, almost dense and weakly dense bases. For the class of weakly dense bases and a special subclass of this class we suggest methods of constructing optimal exact coverings of a graph by bipartite basic graphs and find estimates of the complexity of such coverings. The suggested methods are based on algorithms for optimal exact covering of (0, 1)-matrices by arbitrary (0, 1)-matrices and on the connection between coverings of (0, 1)-matrices and coverings of a graph by bipartite graphs.

Discrete Mathematics and Applications, Walter de Gruyter

Print ISSN: 0924-9266
Volume: 14, 08/2004
Pages: 385 - 407

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