  Using graphs and hypergraphs to systematically model collections of arbitrary
subsets of parties representing {\it ensembles (or collections)} of shared
multipartite CAT states, we study transformations between such {\it ensembles}
under {\it local operations and classical communication (LOCC)}. We show using
partial entropic criteria, that any two such distinct ensembles represented by
{\it $r$-uniform hypergraphs} with the same number of hyperedges (CAT states),
are LOCC incomparable for even integers $r\geq 2$, generalizing results in
\cite{mscthesis,sin:pal:kum:sri}. We show that the cardinality of the largest
set of mutually LOCC incomparable ensembles represented by $r$-uniform
hypergraphs for even $r\geq 2$, is exponential in the number of parties. We
also demonstrate LOCC incomparability between two ensembles represented by
3-uniform hypergraphs where partial entropic criteria do not help in
establishing incomparability. Further we characterize LOCC comparability of EPR
graphs in a model where LOCC is restricted to teleportation and edge
destruction. We show that this model is equivalent to one in which LOCC
transformations are carried out through a sequence of operations where each
operation adds at most one new EPR pair.
