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Planar decompositions and the crossing number of graphs with an excluded minor
 IN GRAPH DRAWING 2006; LECTURE NOTES IN COMPUTER SCIENCE 4372
, 2007
"... Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar ..."
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Cited by 14 (1 self)
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Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar decomposition with bounded width and a linear number of bags. The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. We prove that planar decompositions are intimately related to the crossing number. In particular, a graph with bounded degree has linear crossing number if and only if it has a planar decomposition with bounded width and linear order. It follows from the above result about planar decompositions that every graph with bounded degree and an excluded minor has linear crossing number. Analogous results are proved for the convex and rectilinear crossing numbers. In particular, every graph with bounded degree and bounded treewidth has linear convex crossing number, and every K3,3minorfree graph with bounded degree has linear rectilinear crossing number.
Graphs that Admit Right Angle Crossing Drawings
"... We consider right angle crossing (RAC) drawings of graphs in which the edges are represented by polygonal arcs and any two edges can cross only at a right angle. We show that if a graph with n vertices admits a RAC drawing with at most 1 bend or 2 bends per edge, then the number of edges is at most ..."
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Cited by 6 (1 self)
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We consider right angle crossing (RAC) drawings of graphs in which the edges are represented by polygonal arcs and any two edges can cross only at a right angle. We show that if a graph with n vertices admits a RAC drawing with at most 1 bend or 2 bends per edge, then the number of edges is at most 6.5n and 74.2n, respectively. This is a strengthening of a recent result of Didimo et al. 1
On Crossing Numbers of Geometric Proximity Graphs
"... Let P be a set of n points in the plane. A geometric proximity graph on P is a graph where two points are connected by a straightline segment if they satisfy some prescribed proximity rule. We consider four classes of higher order proximity graphs, namely, the knearest neighbor graph, the krelati ..."
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Cited by 3 (2 self)
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Let P be a set of n points in the plane. A geometric proximity graph on P is a graph where two points are connected by a straightline segment if they satisfy some prescribed proximity rule. We consider four classes of higher order proximity graphs, namely, the knearest neighbor graph, the krelative neighborhood graph, the kGabriel graph and the kDelaunay graph. For k = 0 (k = 1 in the case of the knearest neighbor graph) these graphs are plane, but for higher values of k in general they contain crossings. In this paper we provide lower and upper bounds on their minimum and maximum number of crossings. We give general bounds and we also study particular cases that are especially interesting from the viewpoint of applications. These cases include the 1Delaunay graph and the knearest neighbor graph for small values of k.
A Bipartite Strengthening of the Crossing Lemma
"... Abstract. The celebrated Crossing Lemma states that, in every drawing of a graph with n vertices and m ≥ 4n edges there are at least Ω(m 3 /n 2) pairs of crossing edges; or equivalently, there is an edge that crosses Ω(m 2 /n 2) other edges. We strengthen the Crossing Lemma for drawings in which any ..."
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Cited by 1 (1 self)
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Abstract. The celebrated Crossing Lemma states that, in every drawing of a graph with n vertices and m ≥ 4n edges there are at least Ω(m 3 /n 2) pairs of crossing edges; or equivalently, there is an edge that crosses Ω(m 2 /n 2) other edges. We strengthen the Crossing Lemma for drawings in which any two edges cross in at most O(1) points. We prove for every k ∈ N that every graph G with n vertices and m ≥ 3n edges drawn in the plane such that any two edges intersect in at most k points has two disjoint subsets of edges, E1 and E2, each of size at least ckm 2 /n 2, such that every edge in E1 crosses all edges in E2, where ck> 0 only depends on k. This bound is best possible up to the constant ck for every k ∈ N. We also prove that every graph G with n vertices and m ≥ 3n edges drawn in the plane with xmonotone edges has disjoint subsets of edges, E1 and E2, each of size Ω(m 2 /(n 2 polylog n)), such that every edge in E1 crosses all edges in E2. On the other hand, we construct xmonotone drawings of bipartite dense graphs where the largest such subsets E1 and E2 have size O(m 2 /(n 2 log(m/n))). 1
On Crossings in Geometric Proximity Graphs
"... We study the number of crossings among edges of some higher order proximity graphs of the family of the Delaunay graph. That is, given a set P of n points in the Euclidean plane, we give lower and upper bounds on the minimum and the maximum number of crossings that these geometric graphs defined on ..."
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We study the number of crossings among edges of some higher order proximity graphs of the family of the Delaunay graph. That is, given a set P of n points in the Euclidean plane, we give lower and upper bounds on the minimum and the maximum number of crossings that these geometric graphs defined on P have. 1