A directed graph with identified nodes is defined to represent a set of conditional independence (c.i.) statements. It is shown how new c.i. statements can be read from the graph of an influence diagram and results of Howard and Matheson are rigorised and generalized. A new decomposition theorem, analogous to Kiiveri, Speed and Carlin and requiring no positivity condition, is proved. Connections between influence diagrams and Markov field networks are made explicit. Because all results depend on only three properties of c.i., the theorems proved here can be restated as theorems about other structures like second order processes.