We investigate the relationship between the support of the inverses of generalized covariance matrices and the structure of a discrete graphical model. We show that for certain graph structures, the support of the inverse covariance matrix of indicator variables on the vertices of a graph reflects the conditional independence structure of the graph. Our work extends results which were previously established only for multivariate Gaussian distributions, and partially answers an open question about the meaning of the inverse covariance matrix of a non-Gaussian distribution. We propose graph selection methods for a general discrete graphical model with bounded degree based on possibly corrupted observations, and verify our theoretical results via simulations. Along the way, we also establish new results for support recovery in the setting of sparse high-dimensional linear regression based on corrupted and missing observations.