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Topics in Discrete Mathematics pp 179–213Cite as

Hereditary Properties of Ordered Graphs

  • Conference paper

Part of the book series: Algorithms and Combinatorics ((AC,volume 26))

Abstract

An ordered graph is a graph together with a linear order on its vertices. A hereditary property of ordered graphs is a collection of ordered graphs closed under taking order-preserving isomorphisms of the vertex set, and order-preserving induced subgraphs. If P is a hereditary property of ordered graphs, then P n denotes the collection \( \left\{ {G \in \mathcal{P}:V(G) = [n]} \right\} \), and the function \( n \mapsto \left| {\mathcal{P}_n } \right| \) is called the speed of P.

The possible speeds of a hereditary property of labelled graphs have been extensively studied (see [BBW00] and [Bol98] for example), and more recently hereditary properties of other combinatorial structures, such as oriented graphs ([AS00], [BBM06+c]), posets ([BBM06+a], [BGP99]), words ([BB05], [QZ04]) and permutations ([KK03], [MT04]), have also attracted attention. Properties of ordered graphs generalize properties of both labelled graphs and permutations.

In this paper we determine the possible speeds of a hereditary property of ordered graphs, up to the speed 2n−1. In particular, we prove that there exists a jump from polynomial speed to speed F n, the Fibonacci numbers, and that there exists an infinite sequence of subsequent jumps, from p(n)F n,k to F n,k+1 (where p(n) is a polynomial and F n,k are the generalized Fibonacci numbers) converging to 2n−1. Our results generalize a theorem of Kaiser and Klazar [KK03], who proved that the same jumps occur for hereditary properties of permutations.

The first author was supported during this research by OTKA grant T049398 and NSF grant DMS-0302804, the second by NSF grant ITR 0225610, and the third by a Van Vleet Memorial Doctoral Fellowship.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL, 61801, USA

    József Balogh

  2. Department of Mathematical Sciences, The University of Memphis, Memphis, TN, 38152, USA

    Béla Bollobás & Robert Morris

  3. Trinity College, Cambridge, CB2 1TQ, England

    Béla Bollobás

Authors
  1. József Balogh
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  2. Béla Bollobás
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  3. Robert Morris
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Editor information

Editors and Affiliations

  1. Department of Applied Mathematics and Institute for Theoretical Computer Science ITI, Faculty of Mathematics and Physics, Charles University, Prague, Malostranské nám. 25, 118 00, Praha 1, Czech Republic

    Martin Klazar , Jan Kratochvíl , Martin Loebl , Jiří Matoušek  & Pavel Valtr , , ,  & 

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    Robin Thomas

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Balogh, J., Bollobás, B., Morris, R. (2006). Hereditary Properties of Ordered Graphs. In: Klazar, M., Kratochvíl, J., Loebl, M., Matoušek, J., Valtr, P., Thomas, R. (eds) Topics in Discrete Mathematics. Algorithms and Combinatorics, vol 26. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-33700-8_12

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