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

Abstract

A finite, indecomposable partially ordered set is said to be critically indecomposable if, whenever an element is removed, the resulting induced partially ordered set is not indecomposable. The same terminology can be applied to graphs, tournaments, or any other relational structure whose relations are binary and irreflexive. 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 ⪖ 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 ⪖ 5 there are precisely three of order k. In general, for arbitrary irreflexive binary relational structures, we will see that all critical indecomposables fall into one of nine infinite classes. Four of these classes are even—they contain no structures of odd order and for even k ⪖ 6 they each contain (up to a certain type of equivalence) exactly one structure of order k. The five other classes sre odd—they contain no structures of even order and for each odd k ⪖ 5 they each contain exactly one structure of order k. From this characterization of critically indecomposable structures, it will be evident that all indecomposable substructures of critically indecomposable structures are themselves critically indecomposable. Finally, it is proved that every indecomposable structure of order n + 2 ( n ⪖ 5) has an indecomposable substructure of order n.