  We characterise the optimal broadcast rate for a few classes of
pliable-index-coding problems. This is achieved by devising new lower bounds
that utilise the set of absent receivers to construct decoding chains with
skipped messages. This work complements existing works by considering problems
that are not complete-S, i.e., problems considered in this work do not require
that all receivers with a certain side-information cardinality to be either
present or absent from the problem. We show that for a certain class, the set
of receivers is critical in the sense that adding any receiver strictly
increases the broadcast rate.
