What is the furthest graph from a hereditary property?

  • Authors:

  • Noga Alon;Uri Stav

  • Affiliations:

  • Schools of Mathematics and Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel and Institute for Advanced Study, Princeton, New Jer ...;School of Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel

  • Venue:
  • Random Structures & Algorithms
  • Year:
  • 2008

Quantified Score

Hi-index 0.00

Visualization

Abstract

For a graph property P, the edit distance of a graph G from P, denoted EP(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 EP(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) = EP(G(n,p(P))) + o(n2) with high probability. The proofs combine several tools, including strengthened versions of the Szemerédi regularity lemma, properties of random graphs and probabilistic arguments. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008