As a principal transformation in the reduction process, T.P. Kirkman used the removal of bigons: an inverse of a systematical introduction of bigons used for creating families. In this way, every KL collapses into an irreducible diagram: a solid KL. J. Conway used the same idea, and A. Caudron improved it, so his "worlds" correspond to the different "levels of collapsing", connected with the mentioned graph properties.
At first glance, it is not possible to make any conclusion about the number of components from weighted or any other graphs corresponding to KLs. It represents a secondary property of every KL, so in every subdivision of KLs according to the number of components, the original ordering principle that followed from graph-theoretical derivation was lost. This holds even in the case of A. Caudron's paper, where the possibilities for the classification and for some combinatorial results based on partition theory, remained unexplored.