  An instance of the super-stable matching problem with incomplete lists and
ties is an undirected bipartite graph $G = (A \cup B, E)$, with an adjacency
list being a linearly ordered list of ties. Ties are subsets of vertices
equally good for a given vertex. An edge $(x,y) \in E \backslash M$ is a
blocking edge for a matching $M$ if by getting matched to each other neither of
the vertices $x$ and $y$ would become worse off. Thus, there is no disadvantage
if the two vertices would like to match up. A matching $M$ is super-stable if
there is no blocking edge with respect to $M$. It has previously been shown
that super-stable matchings form a distributive lattice and the number of
super-stable matchings can be exponential in the number of vertices. We give
two compact representations of size $O(m)$ that can be used to construct all
super-stable matchings, where $m$ denotes the number of edges in the graph. The
construction of the second representation takes $O(mn)$ time, where $n$ denotes
the number of vertices in the graph, and gives an explicit rotation poset
similar to the rotation poset in the classical stable marriage problem. We also
give a polyhedral characterisation of the set of all super-stable matchings and
prove that the super-stable matching polytope is integral, thus solving an open
problem stated in the book by Gusfield and Irving .
