Hierarchical topological inference on planar disc maps

Abstract

Given a set V and three relations (formula presented) and (formula presented) we wish to ask whether it is possible to draw the elements v {small element of} V each as a closed disc homeomorph Dv in the plane in such a way that (1) Dv and Dw are disjoint for every (v,w) (formula presented), (2) Dv and Dw have disjoint interiors but share a point of their boundaries for every (v,w) (formula presented), and (3) Dv includes Dw as a sub-region for every (formula presented). This problem arises from the study in geographic information systems. The problem is in NP but not known to be NP-hard or polynomial-time solvable. This paper shows that a nontrivial special case of the problem can be solved in almost linear time.