The liberation set in the inverse eigenvalue problem of a graph

The inverse eigenvalue problem of a graph G is the characterizing all lists eigenvalues real symmetric matrices whose off-diagonal pattern prescribed by adjacencies G. strong spectral property powerful tool in this problem, which identifies entries can be perturbed while controlling and preserving eigenvalues. Matrix Liberation Lemma introduced Barrett et al. 2020 advances notion to more general setting. In paper we revisit prove an equivalent statement, that reduces some technical difficulties applying result. We test our method on form M=A⊕B show how new approach supplements results obtained from only. While extending direct sums graphs, discover surprising connection with zero forcing game Cartesian products graphs. Throughout apply resolve selection open cases for six vertices.