Planar Decompositions and the Crossing Number of Graphs with an Excluded Minor

Wood, David R.; Telle, Jan Arne

doi:10.1007/978-3-540-70904-6_16

Cited by 20 publications

(30 citation statements)

References 87 publications

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“…Later on, they proved a similar result for non-orientable surface, too [2]. Another result of this nature estimates planar crossing numbers of H -minor-free graphs [24].…”

mentioning

confidence: 74%

Planar Crossing Numbers of Graphs of Bounded Genus

Djidjev

Vrťo

2012

Discrete Comput Geom

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Pach and Tóth proved that any n-vertex graph of genus g and maximum degree d has a planar crossing number at most c g dn, for a constant c > 1. We improve on this result by decreasing the bound to O(dgn), and also prove that our result is tight within a constant factor. Our proof is constructive and yields an algorithm with time complexity O(dgn). As a consequence of our main result, we show a relation between the planar crossing number and the surface crossing number.A drawing of a graph G in the plane is an injection of the set of the vertices of G onto points of the plane and a mapping of the set of the edges of G onto simple continuous curves such that the endpoints of each edge are mapped onto the endpoints of its image curve. Moreover, no curve should contain an image of a vertex in its inside and no three curves should intersect in the same point, unless it is an endpoint. The planar crossing number (or simply the crossing number) of G, denoted by cr(G), is the minimum number of edge crossings over all drawings of G in the plane.The concept of crossing numbers was introduced [26] more than 50 years ago by Turán. Although there have been scores of results and publications since, because of A preliminary version of this work was presented at ICALP'2006.H.N. Djidjev ( ) Los

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“…Later on, they proved a similar result for non-orientable surface, too [2]. Another result of this nature estimates planar crossing numbers of H -minor-free graphs [24].…”

mentioning

confidence: 74%

Planar Crossing Numbers of Graphs of Bounded Genus

Djidjev

Vrťo

2012

Discrete Comput Geom

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“…Via a result of Dvořák and Norin [15], this implies they have O(k) treewidth. However, Proposition 8.5 of [30] implies that every outer k-planar graph has treewidth at most 3k + 11, i.e., a better bound on the treewidth than applying the result of Dvořák and Norin to our separators. The treewidth 3k + 11 bound also implies a separation number of 3k + 12, but our bound is better.…”

Section: Outer K-planar Graphsmentioning

confidence: 95%

“…General outer k-planar graphs were considered by Binucci et al [7], who (among other results) showed that, for every k, there is a 2-tree which is not outer k-planar. Wood and Telle [30] considered a slight generalization of outer k-planar graphs in their work and showed that these graphs have treewidth O(k).…”

Section: Introductionmentioning

confidence: 99%

Beyond Outerplanarity

Chaplick¹,

Kryven²,

Liotta³

et al. 2018

Lecture Notes in Computer Science

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We study straight-line drawings of graphs where the vertices are placed in convex position in the plane, i.e., convex drawings. We consider two families of graph classes with nice convex drawings: outer k-planar graphs, where each edge is crossed by at most k other edges; and, outer k-quasi-planar graphs where no k edges can mutually cross. We show that the outer k-planar graphs are ( √ 4k + 1 + 1)-degenerate, and consequently that every outer k-planar graph can be ( √ 4k + 1 +2)colored, and this bound is tight. We further show that every outer kplanar graph has a balanced separator of size at most 2k + 3. For each fixed k, these small balanced separators allow us to test outer k-planarity in quasi-polynomial time, i.e., none of these recognition problems are NP-hard unless ETH fails. For the outer k-quasi-planar graphs we discuss the edge-maximal graphs which have been considered previously under different names. We also construct planar 3-trees that are not outer 3-quasi-planar. Finally, we restrict outer k-planar and outer k-quasi-planar drawings to closed drawings, where the vertex sequence on the boundary is a cycle in the graph. For each k, we express closed outer k-planarity and closed outer k-quasi-planarity in extended monadic second-order logic. Thus, since outer k-planar graphs have bounded treewidth, closed outer k-planarity is linear-time testable by Courcelle's Theorem.

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“…The result was also generalized by Wood and Telle to all graph classes with an excluded minor [12,13] (see also [1]). …”

Section: Introductionmentioning

confidence: 92%