DRP Contract: An Algorithm for Determining Graph Properties in Linear time via Reduction Rules

Monadic second order logic is an generalization of first order logic that allows quantification over set variables as well as individual entities but not over functions. A graph reduction system consists of a finite set of labeled graph pairs (reductions) and a finite set of irreducible graphs (reducts). A reduction system is applied to a graph by replacing the left hand side of a reduction with the right hand side, until no further reductions can be applied. If a graph that is isomorphic to one of the reducts results, then the reduction system accepts the graph. Otherwise it does not. As shown in [1], for every property of a graph that can be written down in monadic second order logic, there exists a reduction system and a finite set of reducts. For every reduction system and set of reducts, there exists a linear time algorithm that will determine whether a particular graph can be reduced to one of the reducts via the reduction system. Unfortunately each step of this chain is non-constructive, so while the existence of the algorithm is not in doubt for a particular reduction system, its exact form currently has to be derived via non-algorithmic means.