  We extend the notion of a tree graph to sequences of prime graphs which are
cycles and edges and name these non-chordal graphs hollow trees. These
structures are especially attractive for palindromic Ising models, which mimic
a symmetry of joint Gaussian distributions. We show that for an Ising model all
defining independences are captured by zero partial correlations and
conditional correlations agree with partial correlations within each prime
graph if and only if the model is palindromic and has a hollow-tree structure.
This implies that the strength of dependences can be assessed locally. We use
the results to find a well-fitting general Ising model with hollow-tree
structure for a set of longitudinal data.
