This paper is eligible for the student paper award. We formulate a broadcast problem, where based on their quality of observations, outputs at various receivers are represented on a graph (called "degradation graph"). If receiver Z is a physically degraded version of receiver Y, then node Z is a child of node Y in this graph. This generalization of the classical degraded broadcast channel provides a framework for various situations where at least some information should be available to receivers with partial (or noisier) observations. Upper and lower bounds are obtained on the capacity region. The upper bound is based on auxiliary variables, whose structure is described by the mirror image of the channel's degradation graph. As a special case of our problem, a packet broadcast network is considered.