Tutte polynomials computable in polynomial time

The Tutte polynomial and related polynomials

The Tutte polynomial was defined originally for graphs and extends to more generally to matroids. The many natural combinatorial interpretations of its evaluations and coefficients for graphs then translate to not obviously related combinatorial quantities in other matroids. For example, the Tutte polynomial evaluated at (2, 0) gives the number of acyclic orientations of a graph. Zaslavsky prov...

Polynomial time computable real functions

In this paper, we study computability and complexity of real functions. We extend these notions, already defined for functions over closed intervals or over the real line to functions over particular real open sets and give some results and characterizations, especially for polynomial time computable functions. Our representation of real numbers as sequences of rational numbers allows us to imp...

Graph Polynomials and Their Applications I: The Tutte Polynomial

We begin our exploration of graph polynomials and their applications with the Tutte polynomial, a renown tool for analyzing properties of graphs and networks. This two-variable graph polynomial, due to W. T. Tutte [Tut47,Tut54, Tut67], has the important universal property that essentially any multiplicative graph invariant with a deletion/contraction reduction must be an evaluation of it. These...

On Polynomial Time Computable Numbers

The computable numbers have already appeared in Turing’s paper [6] in which he defined the Turing machine. Turing defined them in real numbers, the definition is naturally extended to complex numbers. Rice showed that the whole complex computable numbers forms an algebraically closed field [4]. The field is strictly larger than the algebraic number fields Q since it contains transcendental numb...

Tutte Polynomials of Flower Graphs