A tight lower bound for primitivity in k-structures
The recently developed theory of k- structures, which constitute a general notion of hypergraph, can give important contributions to the analysis of combinatorial and algorithmic problems on graphs. The basic result of this theory is a decomposition of k-structures which generalizes the modular decomposition for 2-structures. This one gives, as a special case, the modular decomposition of graphs, which is a well known technique that facilitates the solution of a certain number of problems on graphs. But, there is a subclass of k-structures to which only a trivial decomposition can be applyed, the primitive ones.
In this paper we solve for the general case of k-structures a crucial question for the usefulness of the modular decomposition: to give a tight lower bound for the size of the biggest primitive substructures in primitive k-structures. We show that in a primitive k-structure on n elements, for k>2, n−k+1 is such tight lower bound. This result shows that bounds for primitivity in k-structures do not generalize the case k=2.
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