Although graph theorists are still very far away from solving the Reconstruction Problem, in most cases where a class of graphs has been shown to be reconstructible it has turned out that only a few of the graphs in the deck were needed. So it seems that, in many cases at least, there might actually be more than sufficient information in D(G) to determine G uniquely. This has prompted many researchers to study variants of the Reconstruction Problem in which only some of the information in D(G) is given. Of course, these problems are more difficult than the original form of the Reconstruction Problem, so there is little hope of solving them in general. But their study has given rise to interesting problems, some of which use theory developed in earlier chapters. We shall consider in the next three sections three such variants of the Reconstruction Problem.

The endvertex-deck

One of the first classes of graphs that was shown to be reconstructible was trees [120]. This early result was subsequently improved so that it was shown that, for any tree T, the endvertex-deck D1(T) consisting of those subgraphs T − v with v an endvertex is sufficient to reconstruct T. One natural question that arises is therefore whether, given a graph G with a sufficiently large number of endvertices, G is endvertex-reconstructible, that is, reconstructible from its endvertex-deck D1(G). Bryant has shown that this is, in fact, not true. His result is a clever use of Bouwer's Theorem and it requires the following lemma, whose easy proof is left as an exercise.

Lemma 9.1Let F = ℤ2be the binary field and let X be the k-dimensional vector space consisting of all k-tuples of elements of F. Let Γ be the group of

permutations that are linear transformations on X. Let A be a basis of X and let B be a set of k vectors of X whose sum is zero but such that any proper subset of B is linearly independent. Then,

Theorem 9.2 (Bryant)For any positive integer k there exists a graph G with k endvertices such that G is not endvertex-reconstructible.

The class of Cayley digraphs far from exhausts all vertex-transitive digraphs. The same holds even for the undirected case: The smallest example of a non- Cayley vertex-transitive graph is the Petersen graph; we have already seen that this is not a Cayley graph.

In an earlier chapter we discussed the problem of obtaining non-Cayley vertex-transitive graphs, and we have shown one simple, systematic way of obtaining them. Such constructions often lead to very interesting classes of graphs. The size of the automorphism group is generally not a good indicator for deciding whether a vertex-transitive graph is Cayley, in the sense that a graph may be rich with automorphisms and be non-Cayley, or have a small automorphism group and be Cayley. Examples for the former situation are provided by the odd graphs. The latter case arises with GRRs: in this case, we have a paradox because the smallest possible size for Aut (G) guarantees that the graph is Cayley.

The five classes of graphs that we are going to introduce in this chapter (the generalised Petersen graphs, Kneser graphs, metacirculant graphs, quasi- Cayley digraphs and generalised Cayley graphs) are essentially different. Each of these classes contains infinitely many further non-Cayley graphs and digraphs and some are not even vertex-transitive.

While the methods of orbital graphs and double-cosets representations, as described in the previous chapters, enable us to represent all possible vertextransitive graphs, a general classification theorem for them is presently far beyond us.

Special cases of vertex-transitive graphs have been classified. It is known that all those of prime order are Cayley (actually, circulant graphs, that is, Cayley graphs of a cyclic group). A classification result has also been obtained for the next case, that of order equal to the product of two primes. This required the help of the classification theorem for finite simple groups and further deep group-theoretical results. Further details are given in Section 7.3.

In view of these remarks, this chapter is not intended to offer classification theorems, but rather to illustrate several significant classes of graphs having a good symmetry (mostly, vertex-transitivity).

Each of the next sections is devoted to one of these classes, collecting results that are mostly available only in journal articles.

In this chapter we shall define and explore the most basic properties of what are called orbital graphs. We shall see that, analogously to coset graphs, any vertex-transitive graph is a (generalised) orbital graph. Moreover, this description gives us, in principle, a criterion for arc-transitivity. Consideration of a special type of orbital graph will lead us to strongly regular graphs.

Definitions and basic properties

We have seen in Chapter 1 that, given a permutation group (Γ, V), there is a natural induced action (Γ, V × V) on the ordered pairs of elements of V. We shall now study this action in more detail. First, let us henceforth, unless stated otherwise, suppose in this chapter that the action of Γ on V is transitive. Then the orbits of Γ on V × V are called orbitals. One of them is {(u, u) : u ∈ V}, because (Γ, V) is transitive. We denote this orbital by D0 and we call it the trivial orbital. We usually list the orbitals as D0,D1, …,Dr−1. The number r of orbitals of (Γ, V × V) is called the rank of (Γ, V).

The transpose AT of a subset A of V × V (and, in particular, of any orbital) is defined by AT = {(v, u) : (u, v) ∈ A}. We say that a subset A of V × V, and therefore any orbital, is self-paired when A = AT. A self-paired set containing the two arcs (u, v) and (v, u) is considered to be the edge {u, v}. Note that an orbital D is either self-paired or it contains no pair of opposite arcs. In fact, D ∩ D T ≠ Ø implies D = DT.

Obviously the rank r of (Γ, V) satisfies r ≥ 2 unless V has one element. The case r = 2 clearly corresponds to a 2-transitive permutation group, and we have already seen that such groups are not very interesting as automorphism groups of graphs. From our point of view, therefore, the first interesting case occurs when r = 3. We shall look at this particular case in more detail later in this chapter. We also note that, from now on, any orbital mentioned will be assumed to be nontrivial unless explicitly stated otherwise.

We have seen that the full automorphism group of a Cayley graph Cay(Γ, S) can be larger than Γ. If it happens that Aut(Cay(Γ, S)) is in fact equal to Γ, then, by Theorem 3.5, the automorphism group of Cay(Γ, S) acts regularly on V(Cay(Γ, S)). Conversely, if G is a graph whose automorphism group acts regularly on V(G), then G is a Cayley graph. (This follows from the second part of Theorem 3.5 and also from Sabidussi's Theorem because the stabiliser H in that theorem is trivial.)

A graph G whose automorphism group Γ = Aut(G) acts regularly on V(G) is called a graphical regular representation (GRR) of the group Γ. Therefore G is a GRR of its automorphism group Γ if it is vertex-transitive and the stabiliser Γv of any vertex is just the identity.

Note that in this case, by the Orbit-Stabiliser Theorem, |V(G)| = |Γ|. Also, given any two vertices u, v in G, there is one and only one automorphism α such that α(u) = v. Also, G − u has a trivial automorphism group for any vertex u.

Elementary results

Let us first consider the question of what conditions a graph must satisfy in order to be a GRR of its automorphism group. For convenience we shall state in the next theorem the main comments that we made in the introduction to this chapter, noting again that its proof follows from Theorem 3.5 or Sabidussi's Theorem.

Theorem 5.1Let G be a GRR of its automorphism group Γ. Then G is isomorphic to a Cayley graph Cay(Γ, S) for some set S that generates the group Γ.

Not any Cayley graph is, however, a GRR.

Lemma 5.2Let G = Cay(Γ, S) and let ϕ be an automorphism of Γ such that ϕ(S) = S. Then ϕ, as a permutation of V(G) = Γ, is an automorphism of G fixing the vertex 1.

Therefore, if ϕ is nontrivial, then G is not a GRR.

Proof Since ϕ is a group automorphism, ϕ(1) = 1. Also, ϕ is a graph automorphism Because

These results therefore imply that, for G to be a GRR, it (i) must be isomorphic to a Cayley graph Cay(Γ, S) and (ii) no nontrivial automorphism of Γ can fix the set S.

The Reconstruction Problem is considered one of the foremost unsolved problems in graph theory. In this chapter we shall define the two principal variants of this problem, and we shall give some of the most basic results related to these two problems. We shall also try to convey the flavour of what it takes to reconstruct a class of graphs and how this is related to important graph theoretic properties of the class.

A look at some of the surveys on the Reconstruction Problem, for example, [31, 32, 140, 203], will show that a considerable number of papers on this topic have been published. Also, many reconstruction proofs involve ad hoc case-by-case analysis; at this stage of our knowledge of the problem, it seems impossible to simplify these proofs significantly. It is therefore not the aim of the remaining chapters of this book to give anything like an exhaustive treatment of these results. The reader can see a more complete coverage of the Reconstruction Problem by consulting surveys such as the ones cited earlier and then going to the papers themselves where the results were obtained.

What we shall do in this chapter is give those basic definitions, techniques and results which can be presented succinctly and which are often used in most work on the reconstruction of graphs. We shall also discuss in some detail the reconstruction of particular classes of graphs. In the next chapter we shall look at variations of the main Reconstruction Problem that have been formulated, and in the final two chapters we shall present two general techniques that have been developed and that allow a systematic treatment of reconstruction rather than an ad hoc one for particular graph classes. In these four chapters we shall also try to emphasise as much as possible those aspects of the reconstruction of graphs that are related to symmetry properties of graphs, and we shall as much as possible try to present results that use ideas and theorems that we have already presented in earlier chapters.

Definitions

Given a graph G, the collection of its vertex-deleted subgraphs G − v for all v ∈ V(G) is called the deck of G and is denoted by D(G).

This chapter will deal with operations on graphs; these operations will be generically named ‘products’. The word is already in use for several known operations, like the direct product and the cartesian product. Our approach will try to be unifying; that is, we will offer a mathematical definition of what a ‘graph product’ is. All we require for such an operation is (1) to have the vertex-set as the cartesian product of the vertex-sets of the factors involved in the operation, and (2) to have adjacencies within the resulting product depending solely on the adjacencies inside the factors, according to certain given rules.

Our starting point will be the direct product (also often referred to in the literature as the categorical product), whose definition may be considered the simplest, and upon which we shall obtain all the other products as graph-theoretic unions of suitable direct products.

Most of the content of the present chapter is quite well known, but the results concerning the various graph products that we shall present will often have to be rephrased in order to be consistent with the scheme presented earlier.

We shall especially focus on how the connectedness of the product (as well as other matters, such as those concerning paths) is related to that of the factors. We shall also stress the properties of products as binary operations. Incidentally, let us recall that we consider finite graphs only. Accordingly, some refinements that make sense in the infinite case, such as the distinction between ‘weak’ and ‘strong’ products, will not be our concern. Also, the associative property shall be considered only from this finitary viewpoint.

Here is how this chapter is organised. The general definitions that are typical of our approach will be recalled in Section 6.1. The remaining sections will deal with properties of remarkable special cases, namely the direct product (Section 6.2); the cartesian, strong and lexicographic product (Section 6.3); the modular product; and the ‘Knight’ product (Section 6.4).

General products of graphs

We start with the definition of the direct product.