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}.
Geometric reconfigurations consider the following fundamental problem. Given a starting and a final configuration of an object R, determine if R can move from the starting to the final configuration, subject to some set of movement rules. An object can be a set of disks in the plane, or a graph representing a protein, or a robot's arm, for example. Typical movement rules include maintaining connectivity of the object and avoiding collisions or crossings.
In this paper we study the problem where the object is a planar graph1 G. The starting configuration is a drawing of G in the plane with vertices as distinct points and edges as straight-line segments (and possibly many crossings). Our goal is to relocate as few vertices of G as possible in order to remove all the crossings, that is, to reconfigure G to some straight line crossing-free drawing of G. More formally, a geometric graph is a graph whose vertices are distinct points in the plane (not necessarily in general position) and whose edges are straight-line segments between pairs of points. If the underlying combinatorial graph of G belongs to a class of graphs K, then we say that G is a geometric K graph. For example, if K is the class of planar graphs, then G is a geometric planar graph. Where it causes no confusion, we do not distinguish between the geometric graph and its underlying combinatorial graph. Two edges in a geometric graph cross if they intersect at some point other than a common endpoint. A geometric graph with no pair of crossing edges is called crossing-free.
Consider a geometric graph G with vertex set V (G) = {p 1 , . . . , p n }. A crossing-free geometric graph H with vertex set V (H) = {q 1 , . . . , q n } is called an untangling of G if for all i, j ∈ {1, 2, . . . , n}, q i is adjacent to q j in H if and only if p i is adjacent to p j in G. Furthermore, if p i = q i then we say that p i is fixed, otherwise we say that p i is free. If H is an untangling of G with k vertices fixed, then we say that G can be untangled while keeping k vertices fixed. Clearly only geometric planar graphs can be untangled. Moreover, since every planar graph is isomorphic to some crossing-free geometric graph [5,13], trivially every geometric planar graph can be untangled while keeping at least 2 vertices fixed. For a geometric graph G, let fix(G) denote the maximum number of vertices that can be fixed in an untangling of G.
At the 5th Czech-Slovak Symposium on Combinatorics in Prague in 1998, Mamoru Watanabe asked if every geometric cycle (that is, all polygons) can be untangled while keeping at least εn vertices fixed. Pach and Tardos [9] answered that question in the negative by providing an O((n log n) 2/3 ) upper bound on the number of fixed vertices. Furthermore, they proved that every geometric cycle can be untangled while keeping at least √ n vertices fixed.
Pach and Tardos [9] asked if every geometric planar graph can be untangled while keeping n ε vertices fixed, for some ε > 0. In recent work, Spillner and Wolff [11] showed that geometric planar graphs can be untangled while keeping Ω( log n/ log log n) vertices fixed. The best known bound before that was 3 [6]. In Section 4, we answer the question of Pach and Tardos [9] in the affirmative and provide the first polynomial lower bound for untangling geometric planar graphs. Specifically, our main result is that every n-vertex geometric planar graph can be untangled while keeping (n/3) 1/4 vertices fixed.
There has also been considerable interest in untangling specific classes of geometric planar graphs. Spillner and Wolff [11] studied the untangling of geometric outerplanar graphs and showed that they can be untangled while keeping n/3 vertices fixed; and that for every sufficiently large n, there is an n-vertex outerplanar graph that cannot be untangled while keeping more than 2 √ n -1-1 vertices fixed. Thus Θ( √ n) is the tight bound for outerplanar graphs. A n/3 lower bound for trees was shown by Goaoc et al. [6]. The best known upper bound for trees was O((n log n) 2/3 ), which was in fact proved for geometric paths, by Pach and Tardos [9]. In fact, Pach and Tardos [9] prove this upper bound for geometric cycles. However, their method readily applies for geometric paths. We answer a question posed by Spillner and Wolff [11] and close the gap for trees, by showing that for infinitely many values of n, there is a forest of stars that cannot be untangled while keeping more than 3( √ n -1) vertices fixed. This result is proved in Section 5. In addition, in Section 3, we demonstrate that every geometric tree can be untangled while keeping n/2 vertices fixed, thus slightly improving the n/3 lower bound of Goaoc et al. [6]. We conclude the paper with some open problems.
Untangling graphs has also been studied in [8,12]. Goaoc et al. [6] also studied the computational complexity of the related optimization problems and showed various hardness results.
When proving lower bounds, our goal will
This content is AI-processed based on open access ArXiv data.