Classifying Nearly Complete Intersection Ideals Generated in Degree Two

Nearly complete intersection ideals were introduced by A. Boocher and J. Seiner (2018) and defines a special class of monomial ideals in a polynomial ring. These ideals were used to give a lower bound of the total sum of betti numbers that appear a minimal free resolution of a monomial ideal. In this note we give a graph theoretic classification of nearly complete intersection ideals generated in degree two. In doing so, we define a novel graph operation (the inversion) that is motivated by the definition of this new class of ideals.