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Dynamic maps that allow continuous map rotations, for example, on mobile devices, encounter new geometric labeling issues unseen in static maps before. We study the following dynamic map labeling problem: The input is an abstract map consisting of a set $P$ of points in the plane with attached horizontally aligned rectangular labels. While the map with the point set $P$ is rotated, all labels remain horizontally aligned. We are interested in a consistent labeling of $P$ under rotation, i.e., an assignment of a single (possibly empty) active interval of angles for each label that determines its visibility under rotations such that visible labels neither intersect each other (soft conflicts) nor occlude points in $P$ at any rotation angle (hard conflicts). Our goal is to find a consistent labeling that maximizes the number of visible labels integrated over all rotation angles.We first introduce a general model for labeling rotating maps and derive basic geometric properties of consistent solutions. We show NP-hardness of the above optimization problem even for unit-square labels. We then present a constant-factor approximation for this problem based on line stabbing, and refine it further into an efficient polynomial-time approximation scheme (EPTAS).

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 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. 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. (GD 2014). 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.