Achieving Integrity in Geographic Information Systems—Maps and Nested Maps

This article provides a formal data model which allows to establish geometrical-topological integrity of areal objects in a geographical information system (GIS). The data model leads to an automatic tool able to check consistency of a given set of data and to avoid inconsistencies caused by updates of the database. To this end we start from the mathematical notion of a map which provides an irregular tessellation, i.e., a partition of the plane which is non-overlapping and covering. From another perspective, a map is a plane graph with an explicit representation of faces as its atomic areal components. The concept of nested maps extends this standard notion by the specification of a hierarchical structure which aggregates the set of faces. Such aggregations are common in political and administrative structures. Whereas the mathematical notion of a map is familiar in GIS and the base for many tools supporting topological editing, there was a lack of effectively checkable integrity constraints which are correct and complete, i.e., equivalent, for maps. This article provides an axiomatic, effectively checkable characterization of maps which is equivalent to the standard mathematical one, extends it to nested maps and discusses how to use them in order to achieve and maintain integrity in a GIS.