## Abstract

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 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.

Original language | English (US) |
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Pages (from-to) | 87-104 |

Number of pages | 18 |

Journal | Random Structures and Algorithms |

Volume | 33 |

Issue number | 1 |

DOIs | |

State | Published - Aug 2008 |

## All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)
- Computer Graphics and Computer-Aided Design
- Applied Mathematics

## Keywords

- Hereditary graph properties
- Random graphs
- Regularity lemma