Simple extensions of combinatorial structures

Brignall, Robert; Ruškuc, Nik and Vatter, Vincent (2011). Simple extensions of combinatorial structures. Mathematika, 57(02) pp. 193–214.

DOI: https://doi.org/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 best possible.

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