Combinatorial reconstruction and polynomial invariants

In this dissertation, we look at various problems in the fields of Combinatorial Reconstruction and Polynomial Invariants. In particular, in the first half of the dissertation, I solve the Reconstruction Problem for Abelian Groups of arbitrary cardinality and show the plane is 9-reconstructible under orientation-preserving isomorphisms. In the second half of the dissertation, we find the most general invariants of uncoloured and coloured graphs satisfying a contraction-deletion property, prove the conjecture of Arratia. Bollobás and Sorkin regarding the value of the interlace polynomial at −1, and give a state-space definition of the HOMFLY Polynomial, This is all original work. Chapters 1–7 are sole work, whereas Chapter 8 was joint work with Belá Bollobás and Oliver Riordan. Chapter 9 with Paul Balister, Belá Bollobás and Jonathan Cutler and Chapters 10–11 with Belá Bollobás and David Weinreich.