THE RECONSTRUCTION CONJECTURE

Fifty years have elapsed since the Reconstruction Conjecture was proposed, by S.M. Ulam and P.J. Kelly, in 1941. This seems a fitting time, therefore, to report on its status. Because particular aspects of this elusive question have been amply addressed in a recent flurry of expository articles (Ellingham, 1988; Lauri, 1987; Manvel, 1988; Stockmeyer, 1988), we have chosen to present here an overview of the techniques employed, a manual to reconstruction. Each technique will be illustrated and its applications noted, accompanied by appropriate references. For clarity and simplicity, we shall concentrate on the two principal open questions of reconstruction, namely the Reconstruction Conjecture itself and its companion version for edges, the Edge Reconstruction Conjecture, formulated by F. Harary in 1964. Some related questions, amenable to the same or similar techniques, will be discussed briefly at the end of the article. We start with the basic definitions.

1.1 Definitions. A vertex-deleted subgraph of a graph G is a subgraph G – v obtained by deleting a vertex v and its incident edges. The deck of a graph G is the family of (unlabelled) vertex-deleted subgraphs of G; these are the cards of the deck. A reconstruction of a graph G is a graph H with the same deck as G. A graph G is reconstructible if every reconstruction of G is isomorphic to G. Similar definitions apply to digraphs and hypergraphs.

Introduction

Enumeration theory, which aims to count the number of distinct (non-equivalent) elements in a given class of combinatorial objects, constitutes a significant area in combinatorial analysis. The object of constructive enumeration consists of creating a complete list of configurations with given properties [5,8]. There are several reasons which stimulate research in constructive enumeration. Classical methods are not applicable to many interesting classes of objects such as strongly regular graphs, combinatorial designs, error correcting codes, etc. At present, the only available way to count them is by using algorithmic techniques for fixed values of parameters. Lists of objects are important for generating and testing various hypotheses about invariants, characterization, etc. Moreover, examples of designs with given properties are needed in many areas of applied combinatorics such as coding and experiment planning theories, network reliability and cryptography. Algorithms for constructive enumeration frequently require searching in high dimensional spaces and employ sophisticated techniques to identify partial (final) solutions. Such methods may be of independent interest in artificial intelligence, computer vision, neural networks and combinatorial optimization.

There are several common algorithmic approaches which are used to search for combinatorial configurations with particular properties. These can be divided into two broad classes depending on whether or not they search for all possible solutions in a systematic manner. Among the exhaustive techniques backtracking plays a prominent role. A backtrack algorithm attempts to find a solution vector by recursively building up partial solutions one element at a time. The vectors are examined in lexicographical order until all possible candidates for a component have been exhausted after which one backtracks.

INTRODUCTION

The topic of graph perturbations is, like the classical perturbation theory of linear operators [29], concerned primarily with changes in eigenvalues which result from various perturbations. The eigenvalues are those of an adjacency matrix of a graph G, and a perturbation of G is to be thought of as a local modification such as the addition or deletion of a vertex or edge. Here G is a finite undirected graph without loops or multiple edges, and if its vertices are labelled 1,2,…,n then the corresponding adjacency matrix A is (aij) where aij = 1 if vertices i and j are adjacent, and aij = 0 otherwise. The matrix A is regarded as a matrix with real entries, and since A is symmetric, its eigenvalues are real. These eigenvalues are independent of the ordering of the vertices of G, and so we refer to them as the eigenvalues of G. The n eigenvalues together comprise the spectrum of G.

We shall be concerned with three related questions: (1) What algebraic information about G is sufficient to determine the eigenvalues of a given perturbation of G? (2) Does a given eigenvalue increase or decrease under a given perturbation? (3) How can we compare the effects on eigenvalues oi two different perturbations? Such questions were raised in 1979 by Li and Feng [31] in relation to the largest eigenvalue of a graph, and in effect they considered graphs perturbed by the relocation of certain pendant edges.

Interest in Combinatorics continues to grow. For this, the thirteenth British Combinatorial Conference, our mailing list was almost worldwide and contained more than 950 names.

As usual, the British Combinatorial Committee has chosen nine Invited Speakers for the Conference each of whom has been requested to give a lecture of survey type at the Conference and also to prepare a paper for these Proceedings on the subject matter of their talk. Once again, the hoped-for result of a valuable collection of papers to act as a reference source over a wide spectrum of combinatorial topics has been achieved.

In order that this volume should be available for distribution to participants on the first day of the Conference, the usual tight schedule had to be adhered to. The Editor is therefore especially grateful to those who submitted their papers on time and to the referees who almost all carried out their task promptly as well as thoroughly. (At the time this Preface is being written, it is still not certain whether it will be possible to include all nine papers because one or two were either submitted in final form very late and/or were not refereed in time.)

The Rado lecture (which commemorates the late Richard Rado's contribution to British combinatorics) will, on this occasion, be that given by Z. Füredi.

Arrangements have been made for the contributed papers of the Conference to be published in a special issue of Discrete Mathematics and the organizers are grateful to D. R. Woodall for agreeing to act as guest editor for this purpose.

The classical concept of a conic leads in a natural way to the concept of an oval in an arbitrary projective plane: An oval is a subset Ω of points satisfying both of the following properties: i) no three points of Ω are col linear; ii) Q has exactly one 1-secant (also called a tangent) at each one of its points. If the plane is finite and has order n, then an oval consists of n+1 points.

Ovals of finite projective planes have been intensively studied since 1954. The starting point was the famous theorem of B. Segre [94], [95]: In a Desarguesian plane of odd order, the ovals are exactly the irreducible conies.

This paper is a survey of known results in the following areas:

1) The classification problem for ovals in a desarguesian plane of even order.

2) Ovals in finite non desarguesian planes.

3) Pascal's theorem for ovals and abstract ovals.

4) Collineation groups fixing an oval; some characterizations of the finite desarguesian planes.

THE CLASSIFICATION PROBLEM FOR OVALS IN A DESARGUESIAN PLANE OF EVEN ORDER

In 1956 Segre pointed out that his result on the characterization of conies cannot be extended to desarguesian planes of even order. The classification of ovals in these planes is still an open problem and seems to be very complex.

We give a brief account of the known ovals in desarguesian planes of even order, but for detailed information concerning the extensive theory of ovals developed by Segre and his school the reader is referred in particular to the books [53], [98]. Quite recently, some new investigations have been carried out. Details will be found in the survey papers [15], [80].