### Abstract

The classical Stability Theorem of Erdös and Simonovits can be stated as follows. For a monotone graph property P, let t≥2 be such that t+1=min{_{x}(H):H∉P }. Then any graph G*ΕP on n vertices, which was obtained by removing at most (1/t+o(1))(n2) edges from the complete graph G=K_{n}, has edit distance o(n^{2}) to T _{n}(t), the Turán graph on n vertices with t parts. In this paper we extend the above notion of stability to hereditary graph properties. It turns out that to do so the complete graph K_{n} has to be replaced by a random graph. For a hereditary graph property P , consider modifying the edges of a random graph G=G(n,1/2) to obtain a graph G* that satisfies P in (essentially) the most economical way. We obtain necessary and sufficient conditions on P , which guarantee that G* has a unique structure. In such cases, for a pair of integers (r,s), which depends on P , G* , has distance o(n^{2}) to a graph T_{n}(r,s,1/2) almost surely. Here T_{n}(r,s,1/2) denotes a graph, which consists of almost equal-sized r+s parts, r of them induce an independent set, s induce a clique and all the bipartite graphs between parts are quasi-random (with edge density 1/2). In addition, several strengthened versions of this result are shown.

Original language | English (US) |
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Pages (from-to) | 65-83 |

Number of pages | 19 |

Journal | Journal of Graph Theory |

Volume | 62 |

Issue number | 1 |

DOIs | |

State | Published - Sep 2009 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Geometry and Topology

### Keywords

- Cut distance
- Edit distance
- Extremal graph theory
- Hereditary properties
- Stability theorem

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## Cite this

*Journal of Graph Theory*,

*62*(1), 65-83. https://doi.org/10.1002/jgt.20388