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 a traversal of the corresponding curve from s to any point t' on the curve the distance to t' is non-increasing. A path has increasing chords 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. Moreover, we give 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.