A Two-Variable Interlace Polynomial

We introduce a new graph polynomial in two variables. This “interlace” polynomial can be computed in two very different ways. The first is an expansion analogous to the state space expansion of the Tutte polynomial; the significant differences are that our expansion is over vertex rather than edge subsets, and the rank and nullity employed are those of an adjacency matrix rather than an incidence matrix.

The second computation is by a three-term reduction formula involving a graph pivot; the pivot arose previously in the study of interlacement and Euler circuits in four-regular graphs.

We consider a few properties and specializations of the two-variable interlace polynomial. One specialization, the “vertex-nullity interlace polynomial”, is the single-variable interlace graph polynomial we studied previously, closely related to the Tutte–Martin polynomial on isotropic systems previously considered by Bouchet. Another, the “vertex-rank interlace polynomial”, is equally interesting. Yet another specialization of the two-variable polynomial is the independent-set polynomial.