Nesting in Euler Diagrams: syntax, semantics and construction

This paper considers the notion of nesting in Euler diagrams, and how nesting affects the interpretation and construction of such diagrams. After setting up the necessary definitions for concrete Euler diagrams (drawn in the plane) and abstract diagrams (having just formal structure), the notion of nestedness is defined at both concrete and abstract levels. The concept of a dual graph is used to give an alternative condition for a drawable abstract Euler diagram to be nested. The natural progression to the diagram semantics is explored and we present a "nested form" for diagram semantics. We describe how this work supports tool-building for diagrams, and how effective we might expect this support to be in terms of the proportion of nested diagrams.