Outerplanar crossing numbers of planar graphs

Lecture Notes in Computer Science, 2012

The bend-number b(G) of a graph G is the minimum k such that G may be represented as the edge intersection graph of a set of grid paths with at most k bends. We confirm a conjecture of Biedl and Stern showing that the maximum bend-number of outerplanar graphs is 2. Moreover we improve the formerly known lower and upper bound for the maximum bend-number of planar graphs from 2 and 5 to 3 and 4, respectively.

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.

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.

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.

Lecture Notes in Computer Science, 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 min-max formulas involving efficiently computable lower and upper bounds. These min-max 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 2006). On the other hand, we show that it is NP-hard to compute a weighted version of the crossing number for near-planar graphs.

Lecture Notes in Computer Science, 2005

An outerplanar (also called circular, convex, one-page) drawing of an n-vertex graph G is a drawing in which the vertices are placed on a circle and each edge is drawn using one straight-line segment. We derive exact results for the minimal number of crossings in any outerplanar drawings of the following classes of graphs: 3-row meshes, Halin graphs and complete p−partite graphs with equal size partite sets.

Axioms, 2024

The crossing number of a graph is a significant measure that indicates the complexity of the graph and the difficulty of visualizing it. In this paper, we examine the crossing numbers of join products involving the complete graph K 5 with discrete graphs, paths, and cycles. We analyze optimal drawings of K 5 , identify all five non-isomorphic drawings, and address previously hypothesized crossing numbers for K 5 + P n , and K 5 + C n . Through a simplified approach, we first establish cr(K 5 + D n ) and then extend our method to prove the crossing numbers cr(K 5 + P n ) and cr(K 5 + C n ). These results also lead to new hypotheses for cr(W m + S n ) and cr(W m + W n ) involving wheels and stars. Our findings correct previous inaccuracies in the literature and provide a foundation for future research.

Journal of Combinatorial Theory, Series B, 1972

In this paper we obtain a combinatorial lower bound 6,(G) for the crossing number cr,(G) of a graph G in the closed orientable surface of genus g, and we conjecture that equality holds in a wide range of interesting cases. The lower bound is applied to the crossing number of the l-skeleton of a d-dimensional cube to show that this crossing number must be at least 4, and a constructive technique is used to show that the crossing number is at most 8. Finally, we show that the crossing number of any graph is at most k2 times the crossing number of the underlying simple graph, where k = maximum multiplicity of an edge.

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.

Discrete Mathematics, 2004

Proceedings of the twenty-fourth annual symposium on Computational geometry - SCG '08, 2008

The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. Our main result is that every graph G that does not contain a fixed graph as a minor has crossing number O(∆n), where G has n vertices and maximum degree ∆. This dependence on n and ∆ is best possible. This result answers an open question of Wood and Telle [New York J. Mathematics, 2007], who proved the best previous bound of O(∆ 2 n). In addition, we prove that every K5-minor-free graph G has crossing number at most 2 P v deg(v) 2 , which again is the best possible dependence on the degrees of G. We also study the convex and rectilinear crossing numbers, and prove an O(∆n) bound for the convex crossing number of bounded pathwidth graphs, and a P v deg(v) 2 bound for the rectilinear crossing number of K3,3-minor-free graphs.

Journal of Combinatorial Theory, 1970

Several argtmaents are presented which provide restrictions on the possible number of crossings in drawings of bipartite graphs. In particular it is shown that c~(Ks.,,) = 4[~nll 89-1)] and er(K~.~) = 6[ 89189-1)1.

2003

We introduce the Orchard crossing number, which is defined in a similar way to the well-known rectilinear crossing number. We compute the Orchard crossing number for some simple families of graphs. We also prove some properties of this crossing number. Moreover, we define a variant of this crossing number which is tightly connected to the rectilinear crossing number, and compute it for some simple families of graphs.

The electronic journal of combinatorics

We extend the lower bound in [15] for the outerplanar crossing number (in other terminologies also called convex, circular and one-page book crossing number) to a more general setting. In this setting we can show a better lower bound for the outerplanar crossing number of hypercubes than the best lower bound for the planar crossing number. We exhibit further sequences of graphs, whose outerplanar crossing number exceeds by a factor of log n the planar crossing number of the graph. We study the circular arrangement problem, as a lower bound for the linear arrangement problem, in a general fashion. We obtain new lower bounds for the circular arrangement problem. All the results depend on establishing good isoperimetric functions for certain classes of graphs. For several graph families new near-tight isoperimetric functions are established.

2014

The crossing number cr(G) of a graph G is the minimum number of crossings in a nondegenerate planar drawing of G. The rectilinear crossing number cr(G) of G is the minimum number of crossings in a rectilinear nondegenerate planar drawing (with edges as straight line segments) of G. Zarankiewicz proved in 1952 that cr(Kn1,n2) ≤ Z(n1, n2):= n1