Directed labeled graphs (DLGs) provide a good starting point for visual data & knowledge representation but cannot straightforwardly represent non-binary relationships, nested structures, and relation descriptions. These advanced features require encoded constructs with auxiliary nodes and relationships, which also need to be kept separate from straightforward constructs. Therefore, various extensions of DLGs have been proposed for data & knowledge representation, including n-ary relationships

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... s directed labeled hyperarcs, graph partitionings (possibly interfaced as complex nodes), and (hyper)arc labels used as nodes of other (hyper)arcs. Ontologies and rules have used extended logics for knowledge representation such as description logic, object/frame logic, higher-order logic, and modal logic. The paper demonstrates how data & knowledge representation with graphs and logics can be reconciled, inspiring flexible name specification. It proceeds from simple to extended graphs for logics needed in AI and the Semantic Web. Along with its visual introduction, each graph construct is mapped to its corresponding symbolic logic construct. These graph-logic extensions constitute a systematics defined by orthogonal axes, which has led to the Grailog 1.0 language aligned with the Web-rule industry standard RuleML 1.0.