Upward Embeddings and Orientations of Undirected Planar Graphs

An upward embedding of an embedded planar graph specifies, for each vertex v, which edges are incident on v “above” or “below” and, in turn, induces an upward orientation of the edges from bottom to top. In this paper we characterize the set of all upward embeddings and orientations of an embedded planar graph by using a simple flow model, which is related to that described by Bousset [3] to characterize bipolar orientations. We take advantage of such a flow model to compute upward orientations with the minimum number of sources and sinks of 1-connected embedded planar graphs. We finally devise a new algorithm for computing visibility representations of 1-connected planar graphs using our theoretic results. Communicated by Giuseppe Liotta and Ioannis G. Tollis: submitted October 2001; revised April 2002. Research partially supported by “Progetto ALINWEB: Algoritmica per Internet e per il Web”, MIUR Programmi di Ricerca Scientifica di Rilevante Interesse Nazionale. W. Didimo and M. Pizzonia, Upward Embeddings, JGAA, 7(2) 221–241 (2003)222