4 - Defining sets in combinatorics: a survey

Summary

Abstract

In a given class of combinatorial structures there may be many distinct objects with the same parameters. Two questions arise naturally.

• Given two such objects, where and how do they differ?

• How much of an individual object is needed to identify it uniquely?

These questions are obviously related, the first leading to the concept of a trade, and the second to that of a defining set. This survey deals with denning sets in block designs, graphs and some related structures. The corresponding trades in each structure are also discussed briefly.

Introduction

We start with a simple example. A graph G = (V, E) consists of a finite set V of elements called vertices, and a set E of unordered pairs of vertices, called edges. The complete graph on v vertices, K_v, is a graph in which all pairs of distinct vertices constitute edges, so that any graph on v or fewer vertices may be considered as a subgraph of K_v.

If v = 2n, then a one-factor of K_v is a set of n unordered pairs which between them contain each element of V precisely once. A defining set of a one-factor is a subset of its edges which uniquely identifies it. More generally, a perfect matching in a graph G on 2n vertices is a set of n edges incident with each vertex of V.