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Euclidean greedy drawings of trees
 In Proc. 21st European Symposium on Algorithms
, 2013
"... Abstract. Greedy embedding (or drawing) is a simple and efficient strategy to route messages in wireless sensor networks. For each sourcedestination pair of nodes s, t in a greedy embedding there is always a neighbor u of s that is closer to t according to some distance metric. The existence of gre ..."
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Abstract. Greedy embedding (or drawing) is a simple and efficient strategy to route messages in wireless sensor networks. For each sourcedestination pair of nodes s, t in a greedy embedding there is always a neighbor u of s that is closer to t according to some distance metric. The existence of greedy embeddings in the Euclidean plane R2 is known for certain graph classes such as 3connected planar graphs. We completely characterize the trees that admit a greedy embedding in R2. This answers a question by Angelini et al. (Graph Drawing 2009) and is a further step in characterizing the graphs that admit Euclidean greedy embeddings. 1
Increasingchord graphs on point sets
 Graph Drawing (GD’14), volume 8871 of LNCS
, 2014
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On selfapproaching and increasingchord drawings of 3connected planar graphs
 International Symposium on Graph Drawing (GD ’14
, 2014
"... An stpath in a drawing of a graph is selfapproaching if during the traversal of the corresponding curve from s to any point t ′ on the curve the distance to t′ is nonincreasing. A path has increasing chords if it is selfapproaching in both directions. A drawing is selfapproaching (increasingch ..."
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An stpath in a drawing of a graph is selfapproaching if during the traversal of the corresponding curve from s to any point t ′ on the curve the distance to t′ is nonincreasing. A path has increasing chords if it is selfapproaching in both directions. A drawing is selfapproaching (increasingchord) if any pair of vertices is connected by a selfapproaching (increasingchord) path. We study selfapproaching and increasingchord drawings of triangulations and 3connected planar graphs. We show that in the Euclidean plane, triangulations admit increasingchord drawings, and for planar 3trees we can ensure planarity. We prove that strongly monotone (and thus increasingchord) drawings of trees and binary cactuses require exponential resolution in the worst case, answering an open question by Kindermann et al. [13]. Moreover, we provide a binary cactus that does not admit a selfapproaching drawing. Finally, we show that 3connected planar graphs admit increasingchord drawings in the hyperbolic plane and characterize the trees that admit such drawings. 1
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, 2015
"... We tackle the problem of constructing increasingchord graphs spanning point sets. We prove that, for every point set P with n points, there exists an increasingchord planar graph with O(n) Steiner points spanning P. The main intuition behind this result is that Gabriel triangulations are increasi ..."
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We tackle the problem of constructing increasingchord graphs spanning point sets. We prove that, for every point set P with n points, there exists an increasingchord planar graph with O(n) Steiner points spanning P. The main intuition behind this result is that Gabriel triangulations are increasingchord graphs, a fact which might be of independent interest. Further, we prove that, for every convex point set P with n points, there exists an increasingchord graph with O(n logn) edges (and with no Steiner points) spanning P. Submitted: