Upward straight-line embeddings of directed graphs into point sets

Authors:
Carla Binucci;Emilio Di Giacomo;Walter Didimo;Alejandro Estrella-Balderrama;Fabrizio Frati;Stephen G. Kobourov;Giuseppe Liotta
Affiliations:
Dipartimento di Ingegneria Elettronica e dell'Informazione -- Università degli Studi di Perugia, Italy;Dipartimento di Ingegneria Elettronica e dell'Informazione -- Università degli Studi di Perugia, Italy;Dipartimento di Ingegneria Elettronica e dell'Informazione -- Università degli Studi di Perugia, Italy;Department of Computer Science -- University of Arizona, USA;Dipartimento di Informatica e Automazione -- Università di Roma Tre, Italy;AT&T Research Labs., USA;Dipartimento di Ingegneria Elettronica e dell'Informazione -- Università degli Studi di Perugia, Italy
Venue:
Computational Geometry: Theory and Applications
Year:
2010

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Abstract

In this paper we study the problem of computing an upward straight-line embedding of a planar DAG (directed acyclic graph) G into a point set S, i.e. a planar drawing of G such that each vertex is mapped to a point of S, each edge is drawn as a straight-line segment, and all the edges are oriented according to a common direction. In particular, we show that no biconnected DAG admits an upward straight-line embedding into every point set in convex position. We provide a characterization of the family of DAGs that admit an upward straight-line embedding into every convex point set such that the points with the largest and the smallest y-coordinate are consecutive in the convex hull of the point set. We characterize the family of DAGs that contain a Hamiltonian directed path and that admit an upward straight-line embedding into every point set in general position. We also prove that a DAG whose underlying graph is a tree does not always have an upward straight-line embedding into a point set in convex position and we describe how to construct such an embedding for a DAG whose underlying graph is a path. Finally, we give results about the embeddability of some sub-classes of DAGs whose underlying graphs are trees on point set in convex and in general position.