On Self-Approaching And Increasing-Chord Drawings Of 3-Connected Planar Graphs

An st-path in a drawing of a graph is self-approaching if during the traversal of the corresponding curve from s to any point t' on the curve the distance to t' is non increasing. A path is increasing-chord if it is self-approaching in both directions. A drawing is self-approaching (increasing-chord) if any pair of vertices is connected by a self-approaching (increasing-chord) path.We study self-approaching and increasing-chord drawings of triangulations and 3-connected planar graphs. We show that in the Euclidean plane, triangulations admit increasing-chord drawings, and for planar 3-trees we can ensure planarity. We prove that strongly monotone (and thus increasing-chord) drawings of trees and binary cactuses require exponential resolution in the worst case, answering an open question by Kindermann et al. [14]. Moreover, we provide a binary cactus that does not admit a self-approaching drawing. Finally, we show that 3-connected planar graphs admit increasing-chord drawings in the hyperbolic plane and characterize the trees that admit such drawings.