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oai:drops-oai.dagstuhl.de:7209 2024-03-06T10:40:00Z ddc:004 open_access

On Planar Greedy Drawings of 3-Connected Planar Graphs Da Lozzo, Giordano D'Angelo, Anthony Frati, Fabrizio Greedy drawings 3-connectivity planar graphs convex drawings A graph drawing is greedy if, for every ordered pair of vertices (x,y), there is a path from x to y such that the Euclidean distance to y decreases monotonically at every vertex of the path. Greedy drawings support a simple geometric routing scheme, in which any node that has to send a packet to a destination "greedily" forwards the packet to any neighbor that is closer to the destination than itself, according to the Euclidean distance in the drawing. In a greedy drawing such a neighbor always exists and hence this routing scheme is guaranteed to succeed. In 2004 Papadimitriou and Ratajczak stated two conjectures related to greedy drawings. The greedy embedding conjecture states that every 3-connected planar graph admits a greedy drawing. The convex greedy embedding conjecture asserts that every 3-connected planar graph admits a planar greedy drawing in which the faces are delimited by convex polygons. In 2008 the greedy embedding conjecture was settled in the positive by Leighton and Moitra. In this paper we prove that every 3-connected planar graph admits a planar greedy drawing. Apart from being a strengthening of Leighton and Moitra's result, this theorem constitutes a natural intermediate step towards a proof of the convex greedy embedding conjecture. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Giordano Da Lozzo and Anthony D'Angelo and Fabrizio Frati 2017 Is Part Of LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2017.33 urn:nbn:de:0030-drops-72095 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.33 eng https://creativecommons.org/licenses/by/3.0/legalcode