The interlace polynomial of a graph

Abstract

Motivated by circle graphs, and the enumeration of Euler circuits, we define a one-variable ''interlace polynomial'' for any graph. The polynomial satisfies a beautiful and unexpected reduction relation, quite different from the cut and fuse reduction characterizing the Tutte polynomial.It emerges that the interlace graph polynomial may be viewed as a special case of the Martin polynomial of an isotropic system, which underlies its connections with the circuit partition polynomial and the Kauffman brackets of a link diagram. The graph polynomial, in addition to being perhaps more broadly accessible than the Martin polynomial for isotropic systems, also has a two-variable generalization that is unknown for the Martin polynomial. We consider extremal properties of the interlace polynomial, its values for various special graphs, and evaluations which relate to basic graph properties such as the component and independence numbers. r

The Algebra of Gene Assembly in Ciliates

“…6. In a similar way as described in this section, the Martin polynomial corresponds to a graph polynomial called the interlace polynomial [4] (or Tutte-Martin polynomial [8]) in which local and edge complementation play a central role. Interestingly, the well-known Tutte polynomial on the diagonal coincides with the interlace polynomial when restricting to bipartite graphs [2].…”

One and Two-Variable Interlace Polynomials: A Spectral Interpretation

“…. , n − 1}, then g = j∈R I (x j + c j ) is a degree |R I | Boolean function of |R I | variables, c j ∈ GF(2), and a = d + j∈R H x j is a degree-one Boolean function of |R H | variables, with d ∈ GF (2). In this case the transform spectra w.r.t.…”

“…Remark: [3] The interlace polynomial q for the path graph satisfies, for n ≥ 2, q n (z) = q n−1 (z) + zq n−2 (z), with q 1 (z) = 1, q 2 (z) = 2z; for the complete graph, q n (z) = 2 n−1 z.…”

“…The interlace polynomial was introduced by Arratia, Bollobás and Sorkin [2,3], as a variant of Tutte and Tutte-Martin polynomials [6]. They defined the interlace polynomial of a graph G, q(G), by means of a recurrence formula, involving local complementation (LC) of the graph.…”

“…In [4], Arratia, Bollobas and Sorkin defined an extension of the interlace polynomial q, defined by themselves in [3], to a new polynomial q(x, y). Here we propose a similar extension of Q as defined by Aigner and Van der Holst in [1] to a new polynomial Q(x, y).…”