Polytopes of independent sets of relations and their 1-skeleta

We characterize the edges of two classes of $0/1$-polytopes whose vertices encode the of a relation on a finite set. The first class includes poset chain polytopes, the vertex packing polytopes from graph theory, some instances of matroid independence polytopes, as well as newly-defined polytopes whose vertices correspond to noncrossing set partitions. In analogy with matroid basis polytopes, the second class is obtained by considering the independent sets of maximal cardinality.