Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures

A hnite. indecomposabic partially ordered set is said to be critically indecomposahlc if. whenever an clement is removed. the resulting induced partially ordered set is not indccomposabic. The same terminology can be applied to graphs. tournaments, or any other relational structure whose relations are binary and irrcflexive. It will be shown in this paper that critically indecomposable partially ordered sets are rather scarce; indeed. there are none of odd order. there is exactly one of order 4. and for each even k 3 6 there are exactly two of order k. The same applies to graphs. For tournaments, there are none have even order, there is exactly one of order 3, and for each odd k B 5 there are precisely three of order k. In general, for arbitrary irreflexivc binary relational structures, we will XC that ail critical indecomposabics fail into one of nine infinite classes. Four of these classes are even-they contain no structures of odd order and for cvcn k 2 6 they each contain (up to a certain type of equivalence) exactly one structure of order k. The five other classes are odd-they contain no structures of even order and for each odd k 35 they each contain cxactiy one structure of order k. From this characterization of critically indccomposabic strucrures. it will bc evident that ail indccomposabfc substructures of critically indccompos;rbic structures are themselves critically indccomp()sabfc. Finally, it is proved that every indecomposable structure of order 11 + 2 (tt 3 5) has an rndccomposabic substructure of order tt