  Well-graded families, extremal systems and maximum systems (the last two in
the sense of VC-theory and Sauer-Shelah lemma on VC-dimension) are three
important classes of set systems. This paper aims to study the notion of
duality in the context of these classes of set systems and then use the
obtained results for studying graphs. More specifically, we are concerned with
the characterization of the finite set systems which themselves and their dual
systems are both well-graded, extremal or maximum. On the way to this goal, and
maybe also of independent interest, we study the structure of the well-graded
families with the property that the size of the system is not much bigger than
the size of its essential domain, that is, the set of elements of the domain
which are shattered by the system as single element subsets. As another target
of the paper, we use the above results to characterize graphs whose set systems
of open or closed neighbourhoods, cliques or independent sets are well-graded,
extremal or maximum. We clarify the relation of such graphs to the celebrated
half-graphs. Through the paper, we frequently relate our investigations to the
VC-dimension of the systems. Also we use one-inclusion graphs associated to set
systems as an important technical tool.
