A rollercoaster is a sequence of real numbers for which every maximal contiguous subsequence, that is increasing or decreasing, has length at least three. By translating this sequence to a set of points in the plane, a rollercoaster can be defined as a polygonal path for which every maximal subpath, with positive-or negative-slope edges, has at least three points. Given a sequence of distinct real numbers, the rollercoaster problem asks for a maximum-length (not necessarily contiguous) subsequence that is a rollercoaster. It was conjectured that every sequence of n distinct real numbers contains a rollercoaster of length at least n/2 for n > 7, while the best known lower bound is Ω(n/ log n). In this paper we prove this conjecture. Our proof is constructive and implies a linear-time algorithm for computing a rollercoaster of this length. Extending the O(n log n)time algorithm for computing a longest increasing subsequence, we show how to compute a maximum-length rollercoaster within the same time bound. A maximum-length rollercoaster in a permutation of {1,. .. , n} can be computed in O(n log log n) time. The search for rollercoasters was motivated by orthogeodesic point-set embedding of caterpillars. A caterpillar is a tree such that deleting the leaves gives a path, called the spine. A top-view caterpillar is one of degree 4 such that the two leaves adjacent to each vertex lie on opposite sides of the spine. As an application of our result on rollercoasters, we are able to find a planar drawing of every n-node top-view caterpillar on every set of 25 3 n points in the plane, such that each edge is an orthogonal path with one bend. This improves the previous best known upper bound on the number of required points, which is O(n log n). We also show that such a drawing can be obtained in linear time, provided that the points are given in sorted order.

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We study an old geometric optimization problem in the plane. Given a perfect matching M on a set of n points in the plane, we can transform it to a non-crossing perfect matching by a finite sequence of flip operations. The flip operation removes two crossing edges from M and adds two non-crossing edges. Let f (M ) and F (M ) denote the minimum and maximum lengths of a flip sequence on M , respectively. It has been proved by Bonnet and Miltzow (2016) that f (M ) = O(n 2 ) and by van Leeuwen and Schoone (1980) where ∆ is the spread of the point set, which is defined as the ratio between the longest and the shortest pairwise distances. This improves the previous bound if the point set has sublinear spread. For a matching M on n points in convex position we prove that f (M ) = n/2 − 1 and F (M ) = n/2 2 ; these bounds are tight. Any bound on F (·) carries over to the bichromatic setting, while this is not necessarily true for f (·). Let M be a bichromatic matching. The best known upper bound for f (M ) is the same as for F (M ), which is essentially O(n 3 ). We prove that f (M ) n − 2 for points in convex position, and f (M ) = O(n 2 ) for semi-collinear points.The flip operation can also be defined on spanning trees. For a spanning tree T on a convex point set we show that f (T ) = O(n log n).

We consider an extension of the triangular-distance Delaunay graphs (TD-Delaunay) on a set P of points in the plane. In TD-Delaunay, the convex distance is defined by a fixed-oriented equilateral triangle , and there is an edge between two points in P if and only if there is an empty homothet of having the two points on its boundary. We consider higher-order triangular-distance Delaunay graphs, namely k-TD, which contains an edge between two points if the interior of the homothet of having the two points on its boundary contains at most k points of P . We consider the connectivity, Hamiltonicity and perfect-matching admissibility of k-TD. Finally we consider the problem of blocking the edges of k-TD.

Given a point set P and a class C of geometric objects, G C (P) is a geometric graph with vertex set P such that any two vertices p and q are adjacent if and only if there is some C ∈ C containing both p and q but no other points from P. We study G ▽ (P) graphs where ▽ is the class of downward equilateral triangles (ie. equilateral triangles with one of their sides parallel to the x-axis and the corner opposite to this side below that side). For point sets in general position, these graphs have been shown to be equivalent to half-Θ 6 graphs and TD-Delaunay graphs.The main result in our paper is that for point sets P in general position, G ▽ (P) always contains a matching of size at leastand this bound is tight. We also give some structural properties of G (P) graphs, where is the class which contains both upward and downward equilateral triangles. We show that for point sets in general position, the block cut point graph of G (P) is simply a path. Through the equivalence of G (P) graphs with Θ 6 graphs, we also derive that any Θ 6 graph can have at most 5n − 11 edges, for point sets in general position.