RecognizingP3-Structure: A Switching Approach

AP3is a set of three vertices of a graph that induces a chordless path; theP3-structure of a graph is the set of allP3ʹs. Chvلtal asked if there is a polynomial time algorithm to determine whether an arbitrary three-uniform hypergraph is theP3-structure of some graph. The answer is yes: we describe such an algorithm. Our algorithm is based on switching (as defined by Seidel), a characterization of graphs with the sameP3-structure, a sufficient condition forP3-structure uniqueness, and a reduction of a certain partitioning problem to two-satisfiability.