We investigate classes of graphs and posets that admit
decompositions to obtain or disprove finiteness results
for obstruction sets. To do so we develop a theory of
minimal infinite antichains that allows us to
characterize such antichains by means of the set of
elements below it. In particular we show that the
following classes have infinite antichains with resp.
to the induced subgraph/poset relation: interval graphs
and orders, N-free orders, orders with bounded
decomposition width. On the other hand for
orders with bounded decomposition diameter
finiteness of all antichains is shown. As a consequence
those classes with infinite antichains have undecidable
hereditary properties whereas those with finite
antichains have fast algorithms for all such
properties.
