Extensions of barrier sets to nonzero roots of the matching polynomial

In matching theory, barrier sets (also known as Tutte sets) have been studied extensively due to their connection to maximum matchings in a graph. For a root θθ of the matching polynomial, we define θθ-barrier and θθ-extreme sets. We prove a generalized Berge–Tutte formula and give a characterization for the set of all θθ-special vertices in a graph.