On bijections vs. unary functions
A set of finite structures is in Binary NP if it can be characterized by existential second order formulas in which second order quantification is over relations of arity 2. In [DLS95] subclasses of Binary NP were considered, in which the second order quantifiers range only over certain classes of relations. It was shown that many of these subclasses coincide and that all of them can be ordered in a three-level linear hierarchy, the levels of which are represented by bijections, successor relations and unary functions respectively.
Graph Connectivity is expressible by bijections, thereby showing that the two lower levels of the hierarchy coincide;
the set of graphs with exactly as many vertices as arcs is expressible by unary functions but not by bijections. This shows that level 3 is strictly stronger than the other two levels.
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