Extending graph invariants to embedded graphs

The classical approach (developed by B.Bollobas, O.Riordan, and many others) to extending graph invariants to embedded graphs consists in enhancing the invariants by implementing in them information about the embedding of a graph into a surface. The talk will be devoted to another approach, which is based on the Hopf algebra structures on spaces of graphs, chord diagrams modulo 4-term relations, and delta-matroids. The approach proved to be efficient, and it allows one to extend to embedded graphs such invariants as the degeneracy of a graph, the number of signed Hamiltonian cycles, skew characteristic polynomial, and the Stanley symmetrized chromatic polynomial.