Proving properties of the edge elimination polynomial using equivalent graph polynomials

Averbouch, Godlin and Makowsky define the edge elimination polynomial of a graph by a recurrence relation with respect to the deletion, contraction and extraction of an edge. It generalizes some well-known graph polynomials such as the chromatic polynomial and the matching polynomial. By introducing two equivalent graph polynomials, one enumerating subgraphs and the other enumerating colorings, we show that the edge elimination polynomial of a simple graph is reconstructible from its polynomial deck and that it encodes the degree sequence of an arbitrary graph.