Graph Drawing
DOI: 10.1007/978-3-540-70904-6_16
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Planar Decompositions and the Crossing Number of Graphs with an Excluded Minor

Abstract: Abstract. 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 a… Show more

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Cited by 20 publications
(30 citation statements)
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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%
“…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%
“…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%
“…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%