Outerplanar crossing numbers of planar graphs (original) (raw)
On Graph Crossing Number and Edge Planarization
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.
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.
One-and two-page crossing numbers for some types of graphs
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.
Random Structures and Algorithms, 2008
The biplanar crossing number cr 2 (G) of a graph G is min{cr(G 1 )+ cr(G 2 )}, where cr is the planar crossing number and G 1 ∪ G 2 = G. We show that cr 2 (G) ≤ (3/8)cr(G). Using this result recursively, we bound the thickness by Θ(G) − 2 ≤ Kcr 2 (G) .4057 log 2 n with some constant K. A partition realizing this bound for the thickness can be obtained by a polynomial time randomized algorithm. We show that for any size exceeding a certain threshold, there exists a graph G of this size, which simultaneously has the following properties: cr(G) is roughly as large as it can be for any graph of that size, and cr 2 (G) is as small as it can be for any graph of that size. The existence is shown using the probabilistic method.
Biplanar Crossing Numbers I: A Survey of Results and Problems
Bolyai Society Mathematical Studies, 2006
We survey known results and propose open problems on the biplanar crossing number. We study biplanar crossing numbers of speci c families of graphs, in particular, of complete bipartite graphs. We nd a few particular exact values and give general lower and upper bounds for the biplanar crossing number. We nd the exact biplanar crossing number of K 5;q for every q.
Improved upper bounds on the crossing number
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.
The Orchard crossing number of an abstract graph
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.
Outerplanar crossing numbers, the circular arrangement problem and isoperimetric functions
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.
Crossing numbers of complete tripartite and
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