  We investigate the supports of extremal martingale measures with
pre-specified marginals in a two-period setting. First, we establish in full
generality the equivalence between the extremality of a given measure $Q$ and
the denseness in $L^1(Q)$ of a suitable linear subspace, which can be seen in a
financial context as the set of all semi-static trading strategies. Moreover,
when the supports of both marginals are countable, we focus on the slightly
stronger notion of weak exact predictable representation property (henceforth,
WEP) and provide two combinatorial sufficient conditions, called "2-link
property" and "full erasability", on how the points in the supports are linked
to each other for granting extremality. When the support of the first marginal
is a finite set, we give a necessary and sufficient condition for the WEP to
hold in terms of the new concepts of $2$-net and deadlock. Finally, we study
the relation between cycles and extremality.
