Multidimensional Dominance Drawings
Giacomo Ortali (1), Ioannis G. Tollis (2) ((1) University of, Perugia, (2) Computer Science Department, University of Crete, Heraklion,, Crete, Greece, Tom Sawyer Software, Inc. Berkeley, CA, U.S.A.)

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 be a DAG with vertices and edges. Two vertices are incomparable if doesn't reach and vice versa. We denote by \emph{width} of a DAG , , the maximum size of a set of incomparable vertices of . In this paper we present an algorithm that computes a dominance drawing of a DAG G in dimensions, where . The time required by the algorithm is , with a precomputation time of , needed to compute a \emph{compressed transitive closure} of , and extra or time, if we want . 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, , of a DAG …
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Graph Theory Research · graph theory and CDMA systems
