Brignall, Robert; Ruškuc, Nik and Vatter, Vincent
(2011).
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DOI (Digital Object Identifier) Link:  https://doi.org/10.1112/S0025579310001518 

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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, ⌈log_{2}(n + 1)⌉, ⌈(n + 1)/2⌉, ⌈(n + 1)/2⌉, ⌈log_{4}(n + 1)⌉, ⌈log_{3}(n + 1)⌉ and 1, respectively. In each case these bounds are best possible.
Item Type:  Journal Article 

Copyright Holders:  2010 University College London 
ISSN:  20417942 
Keywords:  indecomposable graph; modular decomposition; prime graph; simple permutation; simple poset; simple tournament; substitution decomposition 
Academic Unit/Department:  Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics Faculty of Science, Technology, Engineering and Mathematics (STEM) 
Item ID:  29672 
Depositing User:  Robert Brignall 
Date Deposited:  06 Oct 2011 08:38 
Last Modified:  06 Oct 2016 04:46 
URI:  http://oro.open.ac.uk/id/eprint/29672 
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Simple extensions of combinatorial structures. (deposited 12 Sep 2011 09:54)
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