Stability‐type results for hereditary properties

Alon, Noga; Stav, Uri

doi:10.1002/jgt.20388

Cited by 3 publications

(6 citation statements)

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“…Theorem 4.4. ( [6]) LetĜ be the closest graph in P * C 4 to G = G(n, 1 2 ), i.e.Ĝ satisfies ∆(G,Ĝ) = ∆(G, P * C 4 ). Then, w.h.p.,Ĝ is (1, 1)-colorable.…”

Section: A Methods For Reducing Hereditary Propertiesmentioning

confidence: 99%

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Hardness of edge-modification problems

Alon

Stav

2009

Theoretical Computer Science

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For a graph property P consider the following computational problem. Given an input graph G, what is the minimum number of edge modifications (additions and/or deletions) that one has to apply to G in order to turn it into a graph that satisfies P? Namely, what is the edit distance ∆(G, P) of a graph G from satisfying P. Clearly, the computational complexity of such a problem strongly depends on P. For over 30 years this family of computational problems has been studied in several contexts and various algorithms, as well as hardness results, were obtained for specific graph properties. Alon, Shapira and Sudakov studied in [3] the approximability of the computational problem for the family of monotone graph properties, namely properties that are closed under removal of edges and vertices. They describe an efficient algorithm that achieves an o(n 2) additive approximation to ∆(G, P) for any monotone property P, where G is an n-vertex input graph, and show that the problem of achieving an O(n 2−ε) additive approximation is N P-hard for most monotone proeprties. The methods in [3] also provide a polynomial time approximation algorithm which computes ∆(G, P)±o(n 2) for the broader family of hereditary graph properties (which are closed under removal of vertices). In this work we introduce two approaches for showing that improving upon the additive approximation achieved by this algorithm is N Phard for several sub-families of hereditary properties. In addition, we state a conjecture on the hardness of computing the edit distance from being induced H-free for any forbidden graph H.

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“…Theorem 4.4. ( [6]) LetĜ be the closest graph in P * C 4 to G = G(n, 1 2 ), i.e.Ĝ satisfies ∆(G,Ĝ) = ∆(G, P * C 4 ). Then, w.h.p.,Ĝ is (1, 1)-colorable.…”

Section: A Methods For Reducing Hereditary Propertiesmentioning

confidence: 99%

“…On the other hand, it is observed in [6] that in some cases an extremal result of this type is not possible. Namely, for some natural graphs H, there might be two graphs with a very different structure which are essentially the closest graphs in P * H to G(n, 1 2 ).…”

Section: The Limits Of This Approachmentioning

confidence: 99%

Hardness of edge-modification problems

Alon

Stav

2009

Theoretical Computer Science

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“…The CRGs K (1) , K (3) and K (4) give an upper bound on ed Forb (H 9 )(p) of min p 3 , p 1+4p , 1−p 2 .…”

Section: Forb(h 9 )mentioning

confidence: 99%

“…Some results on the edit distance function can be found in a variety of papers [14,6,7,2,3,4,5,11,13]. Much of the background to this paper can be found in a paper by Balogh and the author [8].…”

Section: Introductionmentioning

confidence: 99%

On the computation of edit distance functions

Martin

2015

Discrete Mathematics

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The edit distance between two graphs on the same labeled vertex set is the size of the symmetric difference of the edge sets. The edit distance function of the hereditary property, $\mathcal{H}$, is a function of $p\in[0,1]$ and is the limit of the maximum normalized distance between a graph of density $p$ and $\mathcal{H}$. This paper uses the symmetrization method of Sidorenko in order to compute the edit distance function of various hereditary properties. For any graph $H$, ${\rm Forb}(H)$ denotes the property of not having an induced copy of $H$. We compute the edit distance function for ${\rm Forb}(H)$, where $H$ is any split graph, and the graph $H_9$, a graph first used to describe the difficulties in computing the edit distance function.Comment: 25 pages, 2 figure

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“…For background on the edit distance function, applications thereof and theoretical background, we direct the reader to Balogh and the author [8], Alon and Stav [2,3,4,5], Axenovich, Kézdy and the author [6], and Axenovich and the author [7]. The theoretical background upon which this is based can be traced to papers by Prömel and Steger [15,16,17], Bollobás and Thomason [9,10] and Alekseev [1], among others.…”

Section: Forb(h)mentioning

confidence: 99%

The Edit Distance Function and Symmetrization

Martin

2013

Electron. J. Combin.

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The edit distance between two graphs on the same labeled vertex set is the size of the symmetric difference of the edge sets. The distance between a graph, G, and a hereditary property, H, is the minimum of the distance between G and each G ∈ H. The edit distance function of H is a function of p ∈ [0, 1] and is the limit of the maximum normalized distance between a graph of density p and H.This paper utilizes a method due to Sidorenko [Combinatorica 13(1), pp. 109-120], called "symmetrization", for computing the edit distance function of various hereditary properties. For any graph H, Forb(H) denotes the property of not having an induced copy of H. This paper gives some results regarding estimation of the function for an arbitrary hereditary property. This paper also gives the edit distance function for Forb(H), where H is a cycle on 9 or fewer vertices.

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Stability‐type results for hereditary properties

Cited by 3 publications

References 18 publications

Hardness of edge-modification problems

Hardness of edge-modification problems

On the computation of edit distance functions

The Edit Distance Function and Symmetrization

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