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We show that the maximum number of different sizes of maximal complete k-partite induced subgraphs of a graph G of sufficiently large order n is at least (n−k+1)−log2(n−k+1)−4 and at most n−⌊log2(nk!)⌋. We analyze some features of the structure of graphs with the maximum possible number of different sizes of maximal independent sets, and give best-possible estimates for K1,k-free graphs and regular...
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