Multidimensional Dominance Drawings

TL;DR

This paper introduces an efficient algorithm for computing multidimensional dominance drawings of DAGs, providing tighter bounds on their dominance dimension and introducing new concepts like transitive modules and dimensional neck.

Contribution

It presents a novel algorithm that computes dominance drawings in k dimensions with improved bounds, and introduces new concepts to analyze DAGs' structure.

Findings

Algorithm computes dominance drawings in O(kn) time.

Provides tighter bounds on DAG dominance dimension.

Introduces transitive module and dimensional neck concepts.

Abstract

Let $G$ be a DAG with $n$ vertices and $m$ edges. Two vertices $u,v$ are incomparable if $u$ doesn't reach $v$ and vice versa. We denote by \emph{width} of a DAG $G$, $w_G$, the maximum size of a set of incomparable vertices of $G$. In this paper we present an algorithm that computes a dominance drawing of a DAG G in $k$ dimensions, where $w_G \le k \le \frac{n}{2}$. The time required by the algorithm is $O(kn)$, with a precomputation time of $O(km)$, needed to compute a \emph{compressed transitive closure} of $G$, and extra $O(n^2w_G)$ or $O(n^3)$ time, if we want $k=w_G$. Our algorithm gives a tighter bound to the dominance dimension of a DAG. As corollaries, a new family of graphs having a 2-dimensional dominance drawing and a new upper bound to the dimension of a partial order are obtained. We also introduce the concept of transitive module and dimensional neck, $w_N$, of a DAG $G$ and we show how to improve the results given previously using these concepts.