The edge-subgraph poset of a graph and the edge reconstruction conjecture

We consider only finite simple graphs in this paper. Earlier we showed that many invariants of a graph can be computed from the isomorphism class of its partially ordered set of distinct unlabeled non-empty induced subgraphs, that is, the subgraphs themselves are not required. In this paper, we consider an analogous problem of reconstructing an arbitrary graph up to isomorphism from its abstract edge-subgraph poset , which we call the -reconstruction problem. We present an infinite family of graphs that are not -reconstructible and show that the edge reconstruction conjecture is true if and only if the graphs in the family are the only graphs that are not -reconstructible. Let be the set of all unlabeled graphs. Let denote the number of homomorphisms from to . Let be a bijection such that for all , we have . We conjecture that is the identity map. Our conjecture is motivated by the homomorphism cancellation results of Lovász. We prove that the conjecture stated above is weaker than the edge reconstruction conjecture.