Vincent Kusters

Arc Diagrams, Flip Distances, and Hamiltonian Triangulations

We show that every triangulation (maximal planar graph) on n\ge 6 vertices can be flipped into a Hamiltonian triangulation using a sequence of less than n/2 combinatorial edge flips. The previously best upper bound uses 4-connectivity as a means to establish Hamiltonicity. But in general about 3n/5 flips are necessary to reach a 4-connected triangulation. Our result improves the upper bound on the diameter of the flip graph of combinatorial triangulations on n vertices from 5.2n-33.6 to 5n-23. We also show that for every triangulation on n vertices there is a simultaneous flip of less than 2n/3 edges to a 4-connected triangulation. The bound on the number of edges is tight, up to an additive constant. As another application we show that every planar graph on n vertices admits an arc diagram with less than n/2 biarcs, that is, after subdividing less than n/2 (of potentially 3n-6) edges the resulting graph admits a 2-page book embedding.

Column Planarity and Partial Simultaneous Geometric Embedding

We introduce the notion of column planarity of a subset R of the vertices of a graph G. Informally, we say that R is column planar in G if we can assign x-coordinates to the vertices in R such that any assignment of y-coordinates to them produces a partial embedding that can be completed to a plane straight-line drawing of G. Column planarity is both a relaxation and a strengthening of unlabeled level planarity. We prove near tight bounds for column planar subsets of trees: any tree on n vertices contains a column planar set of size at least 14n/17 and for any e?>?0 and any sufficiently large n, there exists an n-vertex tree in which every column planar subset has size at most (5/6?+?e)n. We also consider a relaxation of simultaneous geometric embedding (SGE), which we call partial SGE (PSGE). A PSGE of two graphs G 1 and G 2 allows some of their vertices to map to two different points in the plane. We show how to use column planar subsets to construct k-PSGEs in which k vertices are still mapped to the same point. In particular, we show that any two trees on n vertices admit an 11n/17-PSGE, two outerpaths admit an n/4-PSGE, and an outerpath and a tree admit a 11n/34-PSGE.

The Complexity of Simultaneous Geometric Graph Embedding

Given a collection of planar graphs G1;:::;Gk on the same set V of n vertices, the simultaneous geometric embedding (with mapping) problem, or simply k-SGE, is to nd a set P of n points in the plane and a bijection ’ : V ! P such that the induced straight-line drawings of G1;:::;Gk under ’ are all plane. This problem is polynomial-time equivalent to weak rectilinear realizability of abstract topological graphs, which Kyn

Reconstructing Point Set Order Typesfrom Radial Orderings

We consider the problem of reconstructing the combinatorial structure of a set of \(n\) points in the plane given partial information on the relative position of the points. This partial information consists of the radial ordering, for each of the \(n\) points, of the \(n-1\) other points around it. We show that this information is sufficient to reconstruct the chirotope, or labeled order type, of the point set, provided its convex hull has size at least four. Otherwise, we show that there can be as many as \(n-1\) distinct chirotopes that are compatible with the partial information, and this bound is tight. Our proofs yield polynomial-time reconstruction algorithms. These results provide additional theoretical insights on previously studied problems related to robot navigation and visibility-based reconstruction.

An Optimal Algorithm for Reconstructing Point Set Order Types from Radial Orderings

Given a set P of n labeled points in the plane, the radial system of P describes, for each \(p\in P\), the radial ordering of the other points around p. This notion is related to the order type of P, which describes the orientation (clockwise or counterclockwise) of every ordered triple of P. Given only the order type of P, it is easy to reconstruct the radial system of P, but the converse is not true. Aichholzer et al. (Reconstructing Point Set Order Types from Radial Orderings, in Proc. ISAAC 2014) defined T(R) to be the set of order types with radial system R and showed that sometimes \(|T(R)|=n-1\). They give polynomial-time algorithms to compute T(R) when only given R.

Halving Balls in Deterministic Linear Time

Let \({\mathcal{D}}\) be a set of n pairwise disjoint unit balls in ℝ d and P the set of their center points. A hyperplane \({\mathcal{H}}\) is an m-separator for \({\mathcal{D}}\) if each closed halfspace bounded by \({\mathcal{H}}\) contains at least m points from P. This generalizes the notion of halving hyperplanes (n/2-separators). The analogous notion for point sets has been well studied. Separators have various applications, for instance, in divide-and-conquer schemes. In such a scheme any ball that is intersected by the separating hyperplane may still interact with both sides of the partition. Therefore it is desirable that the separating hyperplane intersects a small number of balls only.

Planar packing of binary trees

In the graph packing problem we are given several graphs and have to map them into a single host graph G such that each edge of G is used at most once. Much research has been devoted to the packing of trees, especially to the case where the host graph must be planar. More formally, the problem is: Given any two trees T1 and T2 on n vertices, we want a simple planar graph G on n vertices such that the edges of G can be colored with two colors and the subgraph induced by the edges colored i is isomorphic to Ti, for i∈{1,2}. A clear exception that must be made is the star tree which cannot be packed together with any other tree. But a popular hypothesis states that this is the only exception, and all other pairs of trees admit a planar packing. Previous proof attempts lead to very limited results only, which include a tree and a spider tree, a tree and a caterpillar, two trees of diameter four and two isomorphic trees. We make a step forward and prove the hypothesis for any two binary trees. The proof is algorithmic and yields a linear time algorithm to compute a plane packing, that is, a suitable two-edge-colored host graph along with a planar embedding for it. In addition we can also guarantee several nice geometric properties for the embedding: vertices are embedded equidistantly on the x-axis and edges are embedded as semi-circles.

Column planarity and partially-simultaneous geometric embedding

We introduce the notion of column planarity of a subset R of the vertices of a graph G. Informally, we say that R is column planar in G if we can assign x-coordinates to the vertices in R such that any assignment of y-coordinates to them produces a partial embedding that can be completed to a plane straight-line drawing of G. Column planarity is both a relaxation and a strengthening of unlabeled level planarity. We prove near tight bounds for the maximum size of column planar subsets of trees: every tree on n vertices contains a column planar set of size at least 14n/17 and for any ɛ > 0 and any sufficiently large n, there exists an n-vertex tree in which every column planar subset has size at most (5/6 + ɛ)n. In addition, we show that every outerplanar graph has a column planar set of size at least n/2. We also consider a relaxation of simultaneous geometric embedding (SGE), which we call partially-simultaneous geometric embedding (PSGE). A PSGE of two graphs G1 and G2 allows some of their vertices to map to two different points in the plane. We show how to use column planar subsets to construct k-PSGEs, which are PSGEs in which at least k vertices are mapped to the same point for both graphs. In particular, we show that every two trees on n vertices admit an 11n/17-PSGE and every two outerplanar graphs admit an n/4-PSGE.

Reconstructing Point Set Order Types from Radial Orderings

We consider the problem of reconstructing the combinatorial structure of a set of n ≥ 5 points in the plane given partial information on the relative position of the points. This partial informatio...

Arc diagrams, flip distances, and Hamiltonian triangulations

Abstract We show that every triangulation (maximal planar graph) on n ≥ 6 vertices can be flipped into a Hamiltonian triangulation using a sequence of less than n / 2 combinatorial edge flips. The previously best upper bound uses 4-connectivity as a means to establish Hamiltonicity. But in general about 3 n / 5 flips are necessary to reach a 4-connected triangulation. Our result improves the upper bound on the diameter of the flip graph of combinatorial triangulations on n vertices from 5.2 n − 33.6 to 5 n − 23 . We also show that for every triangulation on n vertices there is a simultaneous flip of less than 2 n / 3 edges to a 4-connected triangulation. The bound on the number of edges is tight, up to an additive constant. As another application we show that every planar graph on n vertices admits an arc diagram with less than n / 2 biarcs, that is, after subdividing less than n / 2 (of potentially 3 n − 6 ) edges the resulting graph admits a 2-page book embedding.