st-Orientations with Few Transitive Edges

TL;DR

This paper investigates the problem of computing st-orientations of undirected graphs with the fewest transitive edges, proving NP-hardness generally, but providing practical ILP solutions for planar graphs that improve graph drawing compactness.

Contribution

It introduces an ILP model for minimal transitive edges in planar graphs and demonstrates its effectiveness in producing more compact graph drawings.

Findings

Optimal solutions significantly reduce transitive edges.

Reducing transitive edges improves drawing compactness.

NP-hardness established for the general problem.

Abstract

The problem of orienting the edges of an undirected graph such that the resulting digraph is acyclic and has a single source s and a single sink t has a long tradition in graph theory and is central to many graph drawing algorithms. Such an orientation is called an st-orientation. We address the problem of computing st-orientations of undirected graphs with the minimum number of transitive edges. We prove that the problem is NP-hard in the general case. For planar graphs we describe an ILP model that is fast in practice. We experimentally show that optimum solutions dramatically reduce the number of transitive edges with respect to unconstrained st-orientations computed via classical st-numbering algorithms. Moreover, focusing on popular graph drawing algorithms that apply an st-orientation as a preliminary step, we show that reducing the number of transitive edges leads to drawings that are much more compact.