Se p 20 06 SUBGRAPH POSETS , PARTITION LATTICES , GRAPH POLYNOMIALS AND RECONSTRUCTION

Two objects of fundamental importance in the study of graph polynomials are the poset of induced subgraphs of a graph and the lattice of its connected partitions. In my earlier paper I showed that several invariants of a graph can be computed from the isomorphism class of its poset of non-empty induced subgraphs, (that is, subgraphs themselves are not required). In this paper I will prove that the (abstract and folded) connected partition lattice of a graph can be constructed from its abstract poset of induced subgraphs. I will also prove that, except when the graph is a star or a disjoint union of edges, the abstract induced subgraph poset of the graph can be constructed from its abstract folded connected partition lattice. The first construction implies that if a graph polynomial has an expansion on the connected partition lattice then it is reconstructible from the isomorphism class of the poset of non-empty induced subgraphs. Examples of such polynomials are the chromatic symmetric function and the symmetric Tutte polynomial. The second construction implies that a tree can be reconstructed from the isomorphism class of its folded connected partition lattice. This is a possible line of attack for Stanley’s question whether the chromatic symmetric function is a complete invariant of trees. I then show that the symmetric Tutte polynomial of a tree can be computed from the chromatic symmetric function of the tree, thus showing that a question of Noble and Welsh is equivalent to Stanley’s question about the chromatic symmetric function of trees. The paper also develops edge reconstruction theory on the edge subgraph poset, and its relation with Lovász’s homomorphism cancellation laws. In particular I present a conjecture generalising Lovász’s homomorphism cancellation laws, and show that it is weaker than the edge reconstruction conjecture. A characterisation of a family of graphs that cannot be constructed from their abstract edge subgraph posets is also presented. Date: 10 September, 2006. 2000 Mathematics Subject Classification. Primary: 05C60, 05E05.