Graph connectivity, monadic NP and built-in relations of moderate degree

It has been conjectured [FSV93] that an existential second-oder formula, in which the second-order quantification is restricted to unary relations (i.e. a Monadic NP formula), cannot express Graph Connectivity even in the presence of arbitrary built-in relations. In this paper it is shown that Graph Connectivity cannot be expressed by Monadic NP formulas in the presence of arbitrary built-in relations of degree n(o(1)). The result is obtained by using a simplified version of a method introduced in [Sch94] that allows the extension of a local winning strategy for Duplicator, one of the two players in Ehrenfeucht games, to a global winning strategy.