Confluent Hasse diagrams

TL;DR

This paper characterizes when transitively reduced digraphs can be drawn confluently with upward diagrams, providing efficient algorithms for constructing minimal-feature drawings for series-parallel partial orders.

Contribution

It establishes a necessary and sufficient condition for confluent upward drawings based on order dimension and offers optimal construction algorithms for certain classes of digraphs.

Findings

Confluent upward drawings exist if and only if the reachability relation has order dimension at most two.

Efficient algorithms produce minimal-feature drawings in quadratic or linear time.

Drawings are optimal regarding the number of confluent junctions used.

Abstract

We show that a transitively reduced digraph has a confluent upward drawing if and only if its reachability relation has order dimension at most two. In this case, we construct a confluent upward drawing with $O(n^2)$ features, in an $O(n) \times O(n)$ grid in $O(n^2)$ time. For the digraphs representing series-parallel partial orders we show how to construct a drawing with $O(n)$ features in an $O(n) \times O(n)$ grid in $O(n)$ time from a series-parallel decomposition of the partial order. Our drawings are optimal in the number of confluent junctions they use.