  We consider 3 (weighted) posets associated with a graph G - the poset P(G) of
distinct induced unlabelled subgraphs, the lattice Omega(G) of distinct
unlabelled graphs induced by connected partitions, and the poset Q(G) of
distinct unlabelled edge-subgraphs. We study these posets given up to
isomorphism, and their relation to the reconstruction conjectures. We show that
when G is not a star or a disjoint union of edges, P(G) and Omega(G) can be
constructed from each other. The result implies that trees are reconstructible
from their abstract bond lattice. We present many results on the reconstruction
questions about the chromatic symmetric function and the symmetric Tutte
polynomial. In particular, we show that the symmetric Tutte polynomial of a
tree can be constructed from its chromatic symmetric function.
  We classify graphs that are not reconstructible from their abstract
edge-subgraph posets, and further show that the families presented here are the
only graphs not Q-reconstructible if and only if the edge reconstruction
conjecture is true.
  Let f be a bijection from the set of all unlabelled graphs to itself such
that for all unlabelled graphs G and H, hom(G,H) = hom(f(G), f(H)). We
conjecture that f is an identity map. We show that this conjecture is weaker
than the edge reconstruction conjecture. Our conjecture is motivated by
homomorphism cancellation results due to Lov\'asz.
