Untangling polygons and graphs

Untangling is a process in which some vertices of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph C_n while keeping at least \Omega(n^{2/3}) vertices fixed. For any graph G, we also present an upper bound on the number of fixed vertices in the worst case. The bound is a function of the number of vertices, maximum degree and diameter of G. One of its consequences is the upper bound O((n log n)^{2/3}) for all 3-vertex-connected planar graphs.Comment: 11 pages, 3 figure

A polynomial bound for untangling geometric planar graphs

To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput. Geom., 2002] asked if every n-vertex geometric planar graph can be untangled while keeping at least n^\epsilon vertices fixed. We answer this question in the affirmative with \epsilon=1/4. The previous best known bound was \Omega((\log n / \log\log n)^{1/2}). We also consider untangling geometric trees. It is known that every n-vertex geometric tree can be untangled while keeping at least (n/3)^{1/2} vertices fixed, while the best upper bound was O(n\log n)^{2/3}. We answer a question of Spillner and Wolff [arXiv:0709.0170 2007] by closing this gap for untangling trees. In particular, we show that for infinitely many values of n, there is an n-vertex geometric tree that cannot be untangled while keeping more than 3(n^{1/2}-1) vertices fixed. Moreover, we improve the lower bound to (n/2)^{1/2}.Comment: 14 pages, 7 figure

Moving Vertices to Make Drawings Plane

A straight-line drawing $\delta$ of a planar graph $G$ need not be plane, but can be made so by moving some of the vertices. Let shift$(G,\delta)$ denote the minimum number of vertices that need to be moved to turn $\delta$ into a plane drawing of $G$. We show that shift$(G,\delta)$ is NP-hard to compute and to approximate, and we give explicit bounds on shift$(G,\delta)$ when $G$ is a tree or a general planar graph. Our hardness results extend to 1BendPointSetEmbeddability, a well-known graph-drawing problem.Comment: This paper has been merged with http://arxiv.org/abs/0709.017

GraphMaps: Browsing Large Graphs as Interactive Maps

Algorithms for laying out large graphs have seen significant progress in the past decade. However, browsing large graphs remains a challenge. Rendering thousands of graphical elements at once often results in a cluttered image, and navigating these elements naively can cause disorientation. To address this challenge we propose a method called GraphMaps, mimicking the browsing experience of online geographic maps. GraphMaps creates a sequence of layers, where each layer refines the previous one. During graph browsing, GraphMaps chooses the layer corresponding to the zoom level, and renders only those entities of the layer that intersect the current viewport. The result is that, regardless of the graph size, the number of entities rendered at each view does not exceed a predefined threshold, yet all graph elements can be explored by the standard zoom and pan operations. GraphMaps preprocesses a graph in such a way that during browsing, the geometry of the entities is stable, and the viewer is responsive. Our case studies indicate that GraphMaps is useful in gaining an overview of a large graph, and also in exploring a graph on a finer level of detail.Comment: submitted to GD 201

Untangling Circular Drawings: Algorithms and Complexity

We consider the problem of untangling a given (non-planar) straight-line circular drawing $\delta_G$ of an outerplanar graph $G=(V, E)$ into a planar straight-line circular drawing by shifting a minimum number of vertices to a new position on the circle. For an outerplanar graph $G$, it is clear that such a crossing-free circular drawing always exists and we define the circular shifting number shift$(\delta_G)$ as the minimum number of vertices that are required to be shifted in order to resolve all crossings of $\delta_G$. We show that the problem Circular Untangling, asking whether shift$(\delta_G) \le K$ for a given integer $K$, is NP-complete. For $n$-vertex outerplanar graphs, we obtain a tight upper bound of shift$(\delta_G) \le n - \lfloor\sqrt{n-2}\rfloor -2$. Based on these results we study Circular Untangling for almost-planar circular drawings, in which a single edge is involved in all the crossings. In this case, we provide a tight upper bound shift$(\delta_G) \le \lfloor \frac{n}{2} \rfloor-1$ and present a constructive polynomial-time algorithm to compute the circular shifting number of almost-planar drawings.Comment: 20 pages, 10 figures, extended version of ISAAC 2021 pape