# A tight lower bound for primitivity in *k*-structures

## Abstract

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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