Nesting of Indecomposable Substructures in k-ary relations

The modular decomposition of a graph or relation has a large number of combinatorial applications. It divides the structure into a set of "prime" induced substructures, which cannot be further decomposed. Recent work on graphs and k-ary relations has focused on the discovery that prime induced substructures are densely nested when they occur. Lower bounds on the "nesting density" of prime substructures in graphs are used heavily in the only known linear-time algorithm for directed graphs. We improve on the previously known lower bounds for k-ary relations, and show that no further improvement is possible.