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Title: Simple extensions of combinatorial structures
Authors: Brignall, R
Ruskuc, Nik
Vatter, V
Keywords: QA Mathematics
Issue Date: Jul-2011
Citation: Brignall , R , Ruskuc , N & Vatter , V 2011 , ' Simple extensions of combinatorial structures ' Mathematika , vol 57 , no. 2 , pp. 193-214 . , 10.1112/S0025579310001518
Abstract: An interval in a combinatorial structure R is a set I of points which are related to every point in R \ I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a simple structure by structures of smaller sizes — this is called the substitution (or modular) decomposition. In this paper we prove several results of the following type: An arbitrary structure S of size n belonging to a class C can be embedded into a simple structure from C by adding at most f (n) elements. We prove such results when C is the class of all tournaments, graphs, permutations, posets, digraphs, oriented graphs and general relational structures containing a relation of arity greater than 2. The function f (n) in these cases is 2, ⌈log2(n + 1)⌉, ⌈(n + 1)/2⌉, ⌈(n + 1)/2⌉, ⌈log4(n + 1)⌉, ⌈log3(n + 1)⌉ and 1, respectively. In each case these bounds are the best possible.
Version: Preprint
Status: Peer reviewed
ISSN: 0025-5793
Type: Journal article
Rights: This is the author's version of the article, which may be different to the published version. The published version copyright (c) University College London 2010 is available from
Appears in Collections:Centre for Interdisciplinary Research in Computational Algebra (CIRCA) Research
University of St Andrews Research
Pure Mathematics Research

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