Crossing and Weighted Crossing Number of Near-Planar Graphs

Algorithmica, 2009

A nonplanar graph G is near-planar if it contains an edge e such that G − e is planar. The problem of determining the crossing number of a near-planar graph is exhibited from different combinatorial viewpoints. On the one hand, we develop minmax formulas involving efficiently computable lower and upper bounds. These minmax results are the first of their kind in the study of crossing numbers and improve the approximation factor for the approximation algorithm given by Hliněný and Salazar (Graph Drawing GD'06). On the other hand, we show that it is NP-hard to compute a weighted version of the crossing number for near-planar graphs.

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, 2011

Given an n-vertex graph G, a drawing of G in the plane is a mapping of its vertices into points of the plane, and its edges into continuous curves, connecting the images of their endpoints. A crossing in such a drawing is a point where two such curves intersect. In the Minimum Crossing Number problem, the goal is to find a drawing of G with minimum number of crossings. The value of the optimal solution, denoted by OPT, is called the graph's crossing number. This is a very basic problem in topological graph theory, that has received a significant amount of attention, but is still poorly understood algorithmically. The best currently known efficient algorithm produces drawings with O(log 2 n)• (n + OPT) crossings on bounded-degree graphs, while only a constant factor hardness of approximation is known. A closely related problem is Minimum Planarization, in which the goal is to remove a minimum-cardinality subset of edges from G, such that the remaining graph is planar. Our main technical result establishes the following connection between the two problems: if we are given a solution of cost k to the Minimum Planarization problem on graph G, then we can efficiently find a drawing of G with at most poly(d) • k • (k + OPT) crossings, where d is the maximum degree in G. This result implies an O(n • poly(d) • log 3/2 n)approximation for Minimum Crossing Number, as well as improved algorithms for special cases of the problem, such as, for example, k-apex and bounded-genus graphs.

SIAM Journal on Computing, 2013

A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every edge is crossed by at most one other edge. We show that it is NP-hard to decide whether a given near-planar graph is 1-planar. The main idea in both reductions is to consider the problem of simultaneously drawing two planar graphs inside a disk, with some of its vertices fixed at the boundary of the disk. This leads to the concept of anchored embedding, which is of independent interest. As an interesting consequence we obtain a new, geometric proof of NP-completeness of the crossing number problem, even when restricted to cubic graphs. This resolves a question of Hliněný.

Computational Geometry, 2015

We initiate the study of the following problem: Given a non-planar graph G and a planar subgraph S of G, does there exist a straight-line drawing Γ of G in the plane such that the edges of S are not crossed in Γ by any edge of G? We give positive and negative results for different kinds of connected spanning subgraphs S of G. Moreover, in order to enlarge the subset of instances that admit a solution, we consider the possibility of bending the edges of G not in S; in this setting we discuss different trade-offs between the number of bends and the required drawing area.

Discrete Applied Mathematics, 2007

The k-planar crossing number of a graph is the minimum number of crossings of its edges over all possible drawings of the graph in k planes. We propose algorithms and methods for k-planar drawings of general graphs together with lower bound techniques. We give exact results for the k-planar crossing number of K 2k+1,q , for k 2. We prove tight bounds for complete graphs. We also study the rectilinear k-planar crossing number.

Lecture Notes in Computer Science, 1995

We give a survey of recent techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem. Our aim is to emphasize the more general resuits or those results which have an algorithmic flavor, including the recent results of the authors.

JURNAL ILMIAH SAINS

K-CROSSING CRITICAL ALMOST PLANAR GRAPHS ABSTRACT A graph is a pair of a non-empty set of vertices and a set of edges. Graphs can be drawn on the plane with or without crossing of its edges. Crossing number of a graph is the minimal number of crossing among all drawings of the graph on the plane. Graphs with crossing number zero are called planar. A graph is crossing critical if deleting any of its edge decreases its crossing number. A graph is called almost planar if deleting one edge makes the graph planar. This research shows graphs, given an integer k ≥ 1, to build an infinite family of crossing critical almost planar graphs having crossing number k. Keywords: Almost planar graph,crossing critical graph. GRAF K-PERPOTONGAN KRITIS HAMPIR PLANAR ABSTRAK Sebuah graf adalah pasangan himpunan tak kosong simpul dan himpunan sisi. Graf dapat digambar pada bidang dengan atau tanpa perpotongan. Angka perpotongan adalah jumlah perpotongan terkecil di antara semua gambar graf pada bida...

Discrete Mathematics

Given a fixed positive integer k, the k-planar local crossing number of a graph G, denoted by lcr k (G), is the minimum positive integer L such that G can be decomposed into k subgraphs, each of which can be drawn in a plane such that no edge is crossed more than L times. In this note, we show that under certain natural restrictions, the ratio lcr k (G)/lcr1(G) is of order 1/k 2 , which is analogous to the result of Pach et al. [15] for the k-planar crossing number cr k (G) (defined as the minimum positive integer C for which there is a k-planar drawing of G with C total edge crossings). As a corollary of our proof we show that, under similar restrictions, one may obtain a k-planar drawing of G with both the total number of edge crossings as well as the maximum number of times any edge is crossed essentially matching the best known bounds. Our proof relies on the crossing number inequality and several probabilistic tools such as concentration of measure and the Lovász local lemma.

Journal of Graph Theory, 2005

The crossing number cr(G) of a simple graph G with n vertices and m edges is the minimum number of edge crossings over all drawings of G on the R 2 plane. The conjecture made by Erdó´s in 1973 that crðGÞ ! Cm 3 =n 2 was proved in 1982 by Leighton with C ¼ 1=100 and this constant was gradually improved to reach the best known value C ¼ 1=31:08 obtained recently by Pach, Radoič ić , Tardos, and Tó th [4] for graphs such that m ! 103n=16. We improve this result with values for the constant in the range 1=31:08 C < 1=15 where C depends on m=n 2. For example, C > 1=25 for graphs with m=n 2 > 0:291 and n > 22, and C > 1=20 for dense graphs with m=n 2 ! 0:485.

The crossing number cr(G) of a graph G = (V, E) is the smallest number of edge crossings over all drawings of G in the plane. For any k ≥ 1, the k-planar crossing number of G, cr k (G), is defined as the minimum of cr(G 1) + cr(G 2) +. .. + cr(G k) over all graphs G 1 , G 2 ,. .. , G k with ∪ k i=1 G i = G. Pach et al. [Computational Geometry: Theory and Applications 68 2-6, (2018)] showed that for every k ≥ 1, we have cr k (G) ≤ 2 k 2 − 1 k 3 cr(G) and that this bound does not remain true if we replace the constant 2 k 2 − 1 k 3 by any number smaller than 1 k 2. We improve the upper bound to 1 k 2 (1 + o(1)) as k → ∞. For the class of bipartite graphs, we show that the best constant is exactly 1 k 2 for every k. The results extend to the rectilinear variant of the k-planar crossing number.