We consider the graphical lasso formulation for estimating a Gaussian graphical model in the high-dimensional setting. This approach entails estimating the inverse covariance matrix under a multivariate normal model by maximizing the 1 -penalized log-likelihood. We present a very simple necessary and sufficient condition that can be used to identify the connected components in the graphical lasso solution. The condition can be employed to determine whether the estimated inverse covariance

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... will be block diagonal, and if so, then to identify the blocks. This in turn can lead to drastic speed improvements, since one can simply apply a standard graphical lasso algorithm to each block separately. Moreover, the necessary and sufficient condition provides insight into the graphical lasso solution: the set of connected nodes at any given tuning parameter value is a superset of the set of connected nodes at any larger tuning parameter value. This article has supplementary material online.