On the entropy values of hereditary classes of graphs

Alekseev, V. E.

doi:10.1515/dma.1993.3.2.191

Cited by 79 publications

(206 citation statements)

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“…(An extension of this result to all hereditary graph classes was obtained independently by Alexeev [1] and by Bollobás and Thomason [7]. For recent work in this area see [4,2,5].)…”

Section: Introductionmentioning

confidence: 73%

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For most graphsH, mostH‐free graphs have a linear homogeneous set

Kang

McDiarmid

Reed

et al. 2013

Random Struct Algorithms

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Erdős and Hajnal conjectured that for every graph H there is a constant ε = ε(H) > 0 such that every graph G that does not have H as an induced subgraph contains a clique or a stable set of order Ω(|V (G)| ε ).The conjecture would be false if we set ε = 1; however, in an asymptotic setting, we obtain this strengthened form of Erdős and Hajnal's conjecture for almost every graph H, and in particular for a large class of graphs H defined by variants of the colouring number.

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“…(An extension of this result to all hereditary graph classes was obtained independently by Alexeev [1] and by Bollobás and Thomason [7]. For recent work in this area see [4,2,5].)…”

Section: Introductionmentioning

confidence: 73%

“…For recent work in this area see [4,2,5].) Recently, an asymptotic version of Conjecture 1.1 was established by Loebl, Reed, Scott, Thomason and Thomassé [22], involving the quantity | Forb(H) n | estimated in (1). We say that a graph H has the asymptotic Erdős-Hajnal property if there exists a constant ε = ε(H) > 0 such that…”

Section: Introductionmentioning

confidence: 99%

For most graphsH, mostH‐free graphs have a linear homogeneous set

Kang

McDiarmid

Reed

et al. 2013

Random Struct Algorithms

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“…Scheinerman and Zito ( [23]) showed that |P n | belongs to one of few possible classes of functions. Several other papers sharpen these results, concentrating on, e.g., sparse hereditary properties (Balogh, Bollobás and Weinreich [8], [9]), dense hereditary properties (Bollobás and Thomason [10], [11] and Alekseev [1]) and properties of the type P * H (Prömel and Steger [20] , [21], [22]). …”

Section: Related Workmentioning

confidence: 99%

What is the furthest graph from a hereditary property?

Alon

Stav

2008

Random Struct Algorithms

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For a graph property P, the edit distance of a graph G from P, denoted E P (G), is the minimum number of edge modifications (additions or deletions) one needs to apply to G in order to turn it into a graph satisfying P. What is the furthest graph on n vertices from P and what is the largest possible edit distance from P? Denote this maximal distance by ed(n, P). This question is motivated by algorithmic edge-modification problems, in which one wishes to find or approximate the value of E P (G) given an input graph G.A monotone graph property is closed under removal of edges and vertices. Trivially, for any monotone property, the largest edit distance is attained by a complete graph. We show that this is a simple instance of a much broader phenomenon. A hereditary graph property is closed under removal of vertices. We prove that for any hereditary graph property P, a random graph with an edge density that depends on P essentially achieves the maximal distance from P, that is: ed(n, P) = E P (G(n, p(P))) + o(n 2 ) with high probability. The proofs combine several tools, including strengthened versions of the Szemerédi Regularity Lemma, properties of random graphs and probabilistic arguments.

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“…by the colouring number theorem [2,9,6,4]. Similar argument holds for graphs G with α-tw(G) ≤ k. Lemma 2.6.…”

Section: 6mentioning

confidence: 72%

Minor-matching hypertree width

Yolov

2018

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

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In this paper we present a new width measure for a tree decomposition, minor-matching hypertree width, µ-tw, for graphs and hypergraphs, such that bounding the width guarantees that set of maximal independent sets has a polynomially-sized restriction to each decomposition bag. The relaxed conditions of the decomposition allow a much wider class of graphs and hypergraphs to have bounded width compared to other tree decompositions. We show that, for fixed k, there are 2n-vertex graphs of minor-matching hypertree width at most k. A number of problems including Maximum Independence Set, k-Colouring, and Homomorphism of uniform hypergraphs permit polynomial-time solutions for hypergraphs with bounded minor-matching hypertree width and bounded rank. We show that for any given k and any graph G, it is possible to construct a decomposition of minor-matching hypertree width at most O(k 3 ), or to prove that µ-tw(G) > k in time nThis is done by presenting a general algorithm for approximating the hypertree width of well-behaved measures, and reducing µ-tw to such measure. The result relating the restriction of the maximal independent sets to a set S with the set of induced matchings intersecting S in graphs, and minor matchings intersecting S in hypergraphs, might be of independent interest.

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On the entropy values of hereditary classes of graphs

Cited by 79 publications

References 1 publication

For most graphsH, mostH‐free graphs have a linear homogeneous set

For most graphsH, mostH‐free graphs have a linear homogeneous set

What is the furthest graph from a hereditary property?

Minor-matching hypertree width

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