Monadic NP and graph minors

Abstract

In our paper, we prove that Graph Connectivity is not in Monadic NP even in the presence of a built-in relation of arbitrary degree that does not have for an arbitrary, but fixed k ≥ 2 ∈ IN the complete graph K k as a minor. We obtain our result by using the method of indiscernibles and giving a winning strategy for the duplicator in the Ajtai-Fagin Ehrenfeucht-Fraïssé Game. The result is afterwards strengthened to arbitrary forbidden minors and to minor-closed classes of binary relations.