Structure and generation of crossing-critical graphs

2020

A graph $G$ is {$k$-crossing-critical} if $cr(G)\ge k$, but $cr(G\setminus e)<k$ for each edge $e\in E(G)$, where $cr(G)$ is the crossing number of $G$. It is known that for any $k$-crossing-critical graph $G$, $cr(G)\le 2.5k+16$ holds, and in particular, if $\delta(G)\ge 4$, then $cr(G)\le 2k+35$ holds, where $\delta(G)$ is the minimum degree of $G$. In this paper, we improve these upper bounds to $2.5k +2.5$ and $2k+8$ respectively. In particular, for any $k$-crossing-critical graph $G$ with $n$ vertices, if $\delta(G)\ge 5$, then $cr(G)\le 2k-\sqrt k/2n+35/6$ holds.

The simplest graph drawing method is that of putting the vertices of a graph on a line (spine) and drawing the edges as half-circles on k half planes (pages). Such drawings are called kpage book drawings and the minimal number of edge crossings in such a drawing is called the k-page crossing number. In a one-page book drawing, all edges are placed on one side of the spine, and in a two-page book drawing all edges are placed either above or below the spine. The one-page and two-page crossing numbers of a graph provide upper bounds for the standard planar crossing. In this paper, we derive the exact one-page crossing numbers for four-row meshes, present a new proof for the one-page crossing numbers of Halin graphs, and derive the exact two-page crossing numbers for circulant graphs Cn(1, n 2). We also give explicit constructions of the optimal drawings for each kind of graphs.

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.

2013

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.

Journal of Combinatorial Theory, Series B, 1974

In drawings (two edges have at most one point in common, either a node or a crossing) of the complete graph K, in the Euclidean plane there occur at most 2n -2 edges without crossings. This was proved by G. Ringel in [l!. Here the minimal number of edges without crossings in drawings of K,, is determined, and for the existence of values between minimum and maximum is asked. He also remarks in [l], that he knows a special D(K,,) in which no edge without crossings occurs. Here we will show that drawings D(K,J of this kind only exist if n >, 8. Moreover we will determine h(n) for the remaining values of n. 299

Discrete & Computational Geometry, 2006

Twenty years ago, Ajtai et al. and, independently, Leighton discovered that the crossing number of any graph with v vertices and e > 4v edges is at least ce 3 /v 2 , where c > 0 is an absolute constant. This result, known as the "Crossing Lemma," has found many important applications in discrete and computational geometry. It is tight up to a multiplicative constant. Here we improve the best known value of the constant by showing that the result holds with c > 1024/31827 > 0.032. The proof has two new ingredients, interesting in their own right. We show that (1) if a graph can be drawn in the plane so that every edge crosses at most three others, then its number of edges cannot exceed 5.5(v − 2); and (2) the crossing number of any graph is at least 7 3 e − 25 3 (v − 2). Both bounds are tight up to an additive constant (the latter one in the range 4v ≤ e ≤ 5v).

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.

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.

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.

Journal of Combinatorial Theory, Series B, 2010

A conjecture of Richter and Salazar about graphs that are critical for a fixed crossing number k is that they have bounded bandwidth. A weaker well-known conjecture of Richter is that their maximum degree is bounded in terms of k. In this note we disprove these conjectures for every k ≥ 171, by providing examples of k-crossing-critical graphs with arbitrarily large maximum degree.