Every joint density generated over the given parent graph satisfies a set of conditional independent statements, called its independence structure, which could be derived from three known criteria. First we introduce a completely graph theoretical approach to simplify those criteria for special types of parent graphs and their combinations. Then we define an edge minimal graph with different types of edge, called summary graph, and an appropriate criterion to preserve the independence structure after marginalizing over and conditioning on disjoint subsets of the node set of the graph. We also derive algorithms to generate the summary graph from the given parent or other summary graphs.
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