SAMS Subject Classification Number: 06

In this talk we begin by giving a brief introduction on closed sets of a graph. We then discuss minors obtained by contracting closed sets of size \(k\) and apply this in the computation of the \(k\)-defect polynomial. We finally, discuss some values of \(k\) for which the \(k\)-defect polynomial of a graph is the zero polynomial and pose a few questions which can be explored further.

Note that, analogous to the chromatic polynomial (the \(0\)-defect polynomial), the \(k\)-defect polynomials count the number of ways possible to colour a graph with \(\lambda\) colours when allowing \(k\) bad edges.