The Extremal Function for Complete Minors

Thomason, Andrew

doi:10.1006/jctb.2000.2013

Cited by 236 publications

(232 citation statements)

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“…Average degree Ω(t √ log t) forces K t as a minor (Kostochka [12], Thomason [20]), and this bound is best possible. For subdivisions, an old conjecture of Mader and Erdős-Hajnal, which was eventually proved by Bollobás and Thomason [3] and independently by Komlós and Szemerédi [11], says that there is a constant c such that every graph with average degree at least ct 2 contains a subdivision of K t , and this is tight apart from the constant c. The corresponding extremal problem for complete graph immersions has been proposed in [6].…”

Section: Conjecture 11mentioning

confidence: 99%

A minimum degree condition forcing complete graph immersion

DeVos¹,

Dvořák

Fox

et al. 2014

Combinatorica

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An immersion of a graph H into a graph G is a one-to-one mapping f : V (H) → V (G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P uv corresponding to edge uv has endpoints f (u) and f (v). The immersion is strong if the paths P uv are internally disjoint from f (V (H)). It is proved that for every positive integer t, every simple graph of minimum degree at least 200t contains a strong immersion of the complete graph K t . For dense graphs one can say even more. If the graph has order n and has 2cn 2 edges, then there is a strong immersion of the complete graph on at least c 2 n vertices in G in which each path P uv is of length 2. As an application of these results, we resolve a problem raised by Paul Seymour by proving that the line graph of every simple graph with average degree d has a clique minor of order at least cd 3/2 , where c > 0 is an absolute constant. For small values of t, 1 ≤ t ≤ 7, every simple graph of minimum degree at least t − 1 contains an immersion of K t (Lescure and Meyniel [13], DeVos et al. [6]). We provide a general class of examples showing that this does not hold when t is large.

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Section: Conjecture 11mentioning

confidence: 99%

A minimum degree condition forcing complete graph immersion

DeVos¹,

Dvořák

Fox

et al. 2014

Combinatorica

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“…Thomason [41] showed that if the average degree of a graph is at least cp √ log p, then the graph contains K p as a minor (and that this bound is tight). His proof is very complicated and it is not clear if it can be turned into a polynomial-time algorithm to actually find such a minor.…”

Section: Finding Complete Topological Minorsmentioning

confidence: 99%

On Brambles, Grid-Like Minors, and Parameterized Intractability of Monadic Second-Order Logic

Kreutzer¹,

Tazari²

2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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Brambles were introduced as the dual notion to treewidth, one of the most central concepts of the graph minor theory of Robertson and Seymour. Recently, Grohe and Marx showed that there are graphs G, in which every bramble of order larger than the square root of the treewidth is of exponential size in |G|. On the positive side, they show the existence of polynomialsized brambles of the order of the square root of the treewidth, up to log factors. We provide the first polynomial time algorithm to construct a bramble in general graphs and achieve this bound, up to log-factors. We use this algorithm to construct grid-like minors, a replacement structure for grid-minors recently introduced by Reed and Wood, in polynomial time. Using the gridlike minors, we introduce the notion of a perfect bramble and an algorithm to find one in polynomial time. Perfect brambles are brambles with a particularly simple structure and they also provide us with a subgraph that has bounded degree and still large treewidth; we use them to obtain a meta-theorem on deciding certain parameterized subgraph-closed problems on general graphs in time singly exponential in the parameter; the only other result with a similar flavor that is known to us is due to Demaine and Hajiaghayi and obtains a doubly-exponential bound on the parameter (albeit, for a more general class of parameterized problems).The second part of our work deals with providing a lower bound to Courcelle's famous theorem from almost two decades ago, stating that every graph property that can be expressed by a sentence in monadic second-order logic (MSO), can be decided by a linear time algorithm on classes of graphs of bounded treewidth. Whereas much work has been done on designing, improving, and applying algorithms on graphs of bounded treewidth, not much is known on the side of lower bounds: what bound on the treewidth of a class of graphs "forbids" polynomial-time parameterized algorithms to decide MSO-sentences? This question has only recently received attention with the first systematic study appearing in [Kreutzer 2009]. Using our results from the first part of our work we can improve on it significantly and establish a strong lower bound for Courcelle's theorem on classes of colored graphs.

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“…Assume that A is no larger than B, that is, A has at most p/2 vertices. By the result in [18], for sufficiently large d, G(A) has a subgraph H which can be contracted into a complete graph with m vertices where m > (3d/8) 1− . That is, H contains m pairwise disjoint connected subgraphs such that any two of these connected subgraphs are joined by at least one edge.…”

Section: The Smallest Number Of Cycles In 3-connected Graphsmentioning

confidence: 99%