Realizability of Rectangular Euler Diagrams

Abstract

Euler diagrams are a tool for the graphical representation of set relations. Due to their simple way of visualizing elements in the sets by geometric containment, they are easily readable by an inexperienced reader. Euler diagrams where the sets are visualized as aligned rectangles are of special interest. In this work, we link the existence of such rectangular Euler diagrams to the order dimension of an associated order relation. For this, we consider Euler diagrams in one and two dimensions. In the one-dimensional case, this correspondence provides us with a polynomial-time algorithm to compute the Euler diagrams, while the two-dimensional case results in an exponential-time algorithm.