  Kendall's Similarity Shape Theory for constellations of N points in the
carrier space $\mathbb{R}^d$ as quotiented by the similarity group was
developed for use in Probability and Statistics. It was subsequently shown to
reside within Mechanics' Shape-and-Scale Theory, in which points are
interpreted as particles, carrier space plays the role of absolute space, and
the Euclidean group is quotiented out. Let us jointly refer to
Shape(-and-Scale) Theory as Relational Theory, and to its reduced configuration
spaces as relational spaces. We now consider a less structured version: the
Topological Relational Theory of `rubber configurations'. This already encodes
some features of the much more diverse Geometrical Relational Theories. In
contrast with the latter's (stratified) manifold relational spaces, the
former's are graphs: much simpler to treat; their edges encode topological
adjacency. We concentrate on Leibniz spaces, corresponding to indistinguishable
points and mirror-image identification. These are moreover the building blocks
of the distinguishable and (where possible) mirror-image distinct cases'
relational spaces. For connected manifold without boundary carrier spaces,
there are just 3 'rubber relationalisms: $\mathbb{R}$, $\mathbb{S}^1$, and a
joint one for all carrier spaces with $d \geq 2$. For $d \geq 2$, rubber
configurations are in 1:1 correspondence with partitions, with $\mathbb{S}^1$
and $\mathbb{R}$ giving successive refinements. We find that generic and
maximal configurations are universally present as cone points, as are binaries
in the first 2 cases. Deconing leaves us with residue graphs containing the
N-specific information. We provide graph-theoretical nontriviality criteria for
which N = 6, 6 and 5 are minimal across these models, and stronger such for
which N = 8, 8 and 6 are minimal, and outline GR topology-change analogue-model
and N-body problem applications.
