Graph reconstruction—a survey

Graph Reconstruction- A Survey

J. A. Bondy

R. L. Hemminger UNIVERSITY OF WATERLOO

VANDERBILT UNIVERSITY

ABSTRACT

The Reconstruction Conjecture asserts that every finite simple undirected graph on three or more vertices is determined, up to isomorphism, by its collection of vertex-deleted subgraphs. This article reviews the progress made on the conjecture since it was first formulated in 1941 and discusses a number of related questions.

The Reconstruction Conjecture is generally regarded as one of the foremost unsol;ed problems in graph theory. Indeed, Harary (1969) has even classified it as a “graphical disease” because of its contagious nature. According to reliable sources, it was discovered in Wisconsin in 1941 by Kelly and Ulam, and claimed its first victim (P. J. Kelly) in 1942.* There are now more than sixty recorded cases, and relapses occur frequently (this article being a case in point). Our purpose here is to describe and analyse the current status of the disease, identify its more interesting variants, and suggest possible remedies.

We shall, €or the most part, use the terminology and notation of Bondy and Murty;? so a graph G has vertex set V(G), edge set E(G) , v (G) vertices and E ( G ) edges. A subgraph of G obtained by deleting a vertex v together with its incident edges will be referred to as a vertex-deleted subgraph and denoted by G, (rather than G - v ) . Figure 1 exhibits the vertex-deleted subgraphs of a graph.

* Kelly‘s doctoral thesis appeared in that year. ‘F J. A. Bondy and U.S.R. Murty, Graph Theory with Applications. Macmillan,

London, and American Elsevier, New York (1976).

Journal of Graph Theory, Vol. 1 (1977) 227-268 Copyright @ 1977 by John Wiley & Sons, Inc. 227

228 JOURNAL OF GRAPH THEORY

1 I 1 I 1

4 4 4 4 4

FlGURE 1 The vertex-deleted subgraphs of a graph

To formulate the conjecture precisely, we introduce two definitions. A reconstruction of a graph G is a graph H such that V ( H ) = V ( G ) and Huz G, for all u E V(G) . We call G reconstructible if every reconstruction of G is isomorphic to G. Not all graphs are reconstructible; for example, K, and 2 K , are reconstructions of one another. The Reconstruction Conjecture asserts that these are the only nonreconstructible graphs.

Reconstruction Conjecture.* least three vertices are reconstructible.

All finite simple undirected graphs with at

It is sometimes helpful (as Harary, 1964, suggests) to imagine a deck of cards on which the vertex-deleted subgraphs of G are drawn, but not labelled. For example, the graph G in Figure 1 has the deck displayed in Figure 2.

Presented with such a deck, it is of course a routine matter (in the finite case) to find some graph which produces that deck. But the problem that confronts the reconstructor is more demanding. He must show that, regardless of the algorithm used, one necessarily ends up with the same

0 FIGURE 2 The deck for the graph of figure 1

This formulation of the Kelly-Ulam conjecture IS due to Harary (1964)

GRAPH RECONSTRUCTION-A SURVEY 229

graph. (Readers willing to expose themselves to the reconstruction disease may do so by tackling the deck in Figure 2.)

If a simple graph is reconstructible, then so is its complement. This observation, due to Kelly (1957), does not appear to make the Recon- struction Conjecture any easier to prove, but is of use in a systematic search for counterexamples. To date, all graphs with at most nine vertices have been checked (see Kelly, 1957; Harary and Palmer, 1965; Stock- meyer, 1976b; McKay, 1977; and Nijenhuis, 1977 and no counterexample has been unearthed. Furthermore, Muller (1976) has shown that almost all graphs are reconstructible. Although these results tend to confirm one’s belief in the conjecture, it must be admitted that there is no really convincing evidence in its favour. What there is, we present in Sec. 1. The Edge Reconstruction Conjecture, on which rather more progress has been made, is discussed in Sec. 2. Subsequent sections are concerned with the corresponding conjectures for digraphs and infinite graphs, and an account of related reconstruction questions. The survey concludes with a list of unsolved problems.

1. VERTEX RECONSTRUCTION

In this section, we confine our attention to graphs which satisfy the hypotheses of the Reconstruction Conjecture, namely finite, simple, undirected graphs with at least three vertices.

Short of proving the conjecture directly, two approaches suggest themselves. One is to reconstruct classes of graphs, in the hope that eventually enough classes will be found to include all graphs. (We call a class 3 reconstructible if each graph in 3 is reconstructible). The other approach is through the reconstruction of parameters. (A parameter, or indeed any function defined on a class 3 of graphs, is reconstructible if, for each graph G in %, it takes the same value on all reconstructions of G). In fact, as we shall see, these two lines of attack are closely related.

The following fundamental lemma is due to Kelly (1957). We shall appeal to it frequently throughout this survey.

Kelly’s lemma. For any two graphs F and G such that v ( F ) < v(G), the number s(F, G) of subgraphs of G isomorphic to F is reconstructible.

Proof. Each subgraph of G isomorphic to F occurs in exactly v(G)- v ( F ) of the subgraphs G,. Therefore


Since the right-hand side of this identity is clearly reconstructible, so, too, is the left-hand side I

Corollary. For any two graphs F and G such that v(F)< v(G), the number of subgraphs of G which are isomorphic to F, and include a given vertex u, is reconstructible.

Proof. This number is just s(F, G)-s(F, G,) I

Taking F = K , in Kelly’s lemma and in the corollary, we find that the number of edges and the degree sequence, respectively, are reconstructible parameters.

It is now easily seen, as noted by Kelly (1957), that regular graphs are reconstructible. Consider a K-regular graph G. Since the degree sequence of G is reconstructible, all reconstructions .of G are k-regular. But it is clear that all k-regular reconstructions of G are isomorphic, since each can be obtained (up to isomorphism) from any G, by adding a vertex and joining it to all the vertices of degree k - 1 in G,. We deduce that all reconstructions of G are isomorphic.

This proof is typical of many on reconstruction in that it splits naturally into two parts, which we shall refer to as “recognizability” and “weak reconstructibility.” A class 3 of graphs is recognizable if, for each graph G in 3, every reconstruction of G is also in 3, and weakly reconstructible, if, for each graph G in 3, all reconstructions of G that are in % are isomorphic to G. Thus a class 3 is reconstructible if and only if it is recognizable and weakly reconstructible.

Kelly (1957) also noted that disconnected graphs are reconstructible. Several proofs of this result are to be found in the literature (Harary, 1964; Bondy, 1968; Greenwell and Hemminger, 1969; Chartrand and Kronk, 1970; Manvel, 1970a, 1976; Chartrand, Kronk and Schuster, 1973). The one by Manvel is very short indeed. The proof that we give here is due to Greenwell and Hemminger (1973) and is based on a simple, but powerful, counting theorem.

Let 9 be a class of graphs (that is, a family of graphs closed under isomorphism), and let F and G be graphs such that F E 9 and s(F, G) > 0. A subgraph of G which belongs to 9 is called an 9-subgraph of G ; a maximal 9-subgraph of G is one which is contained in no other 9- subgraph of G. For instance, when 9 is the class of connected graphs, the maximal 9-subgraphs of G are the components of G. An (F, G)-chain of length n is a sequence (Xo, XI,. . . , X,) of 9-subgraphs of G such that F? X , c XI c * . c X , c G. Two (F, G)-chains are isomorphic if they have the same length and corresponding terms are isomorphic graphs. The rank of F in G is the length of a longest (F, G)-chain.


Counting Theorem. Let % be a recognizable class of graphs, and let 9 be any class of graphs such that, for every G in 9, each %-subgraph of G is (i) vertex-proper; (ii) contained in a unique maximal 9-subgraph of G. Then, for every F in 9 and every G in 3, the number m(F, G) of maximal $-subgraphs of G isomorphic to F is reconstructible.

(This differs somewhat from the one given by Greenwell and Hemminger). By (i) we may assume, without loss of generality, that v ( F ) < v(G). We first show that

Proof.

where the inner sum extends over all nonisomorphic (F, G)-chains

The proof of (1) is by induction on the rank of F. When rankF=O, rn(F, G)=s(F , G) and so (1) clearly holds. Suppose, now, that (1) holds for all graphs F in 9 of rank less than r, and let rank F = r.

(X*7 XI, * * . , X").

By condition (ii) of the hypothesis, we have

where the sum extends over all nonisomorphic 9-subgraphs X of G. We can rewrite this as

In (2) it suffices to consider only those $-subgraphs X for which s(F,X)>O. Since any such X has rank less than r, the induction hypothesis can be applied to each term m(X, G), yielding (1).

Now let H be a reconstruction of G. Since 54 is recognizable, conditions (i) and (ii) of the hypothesis hold, and hence formula (1) applies, when G is replaced by H. Therefore, by Kelly's lemma, the right-hand side of (1) is reconstructible, and so the left-hand side is, too I

We remark in passing that the counting theorem includes Kelly's lemma as a special case: choose $ to be the class of all graphs isomorphic to F and 54 to be the class of all reconstructions of G.


Corollary. Disconnected graphs are reconstructible.

Proof. A graph G is disconnected if and only if at most one G, is connected. Therefore, disconnected graphs are recognizable. The counting theorem, with 9 as the class of connected graphs and % as the class of disconnected graphs, establishes weak reconstructibility I

We turn next to connected graphs, starting with trees. A tree is either central (has one center) or bicentral (has two, adjacent, centers). By making use of this property, Kelly (1957) proved that trees are reconstructible. We shall give a simpler proof by employing the counting theorem.

Theorem. Trees are reconstructible.

Proof. Trees are recognizable, since a graph G is a tree if and only if G is connected and E = v - 1.

A tree is a path if and only if each degree is at most two. Therefore paths are recognizable, and hence reconstructible.

In a tree which is not a path, every longest path is a vertex-proper subgraph. It follows from Kelly’s lemma that the diameter and radius of a tree are reconstructible, and hence that central and bicentral trees are recognizable.

A vertex of a tree is peripheral if it is an end of a longest path. Since u is peripheral if and only if d ( u ) = 1 and u is in a longest path, the number of peripheral vertices is reconstructible.

A branch of a central (bicentral) tree is a maximal subtree in which the central vertex (central edge) is of degree one (is incident with a vertex of degree one). A branch is radial if it includes a peripheral vertex of the tree. Note that a bicentral tree has exactly two branches, both of which are radial. A tree is basic if it has exactly two branches, just one of which is a path; the path branch is the stern and the other branch the top.

Now a tree of radius r (and not a path) is basic if and only if it contains no subgraph of one of the three types shown in Figure 3 (where the centers are indicated in black and the distances a and b range between 1 and r - 1). Trees of these types are easily recognizable. (For example, a tree

L P J - O - J - 0 0 b 0 b

T y p e I Type 2 T y p e 3 ,

FIGURE 3 Non-basic trees.


G is of type 1 if and only if it contains a path of length 2 r = v - 2 and r f 2 paths of length r + l ) . Therefore, by Kelly’s lemma, basic trees are recognizable.

Basic trees are also weakly reconstructible, For let G be a central (bicentral) basic tree. Then all reconstructions of G are isomorphic, since each can be obtained, up to isomorphism, from the bicentral (central) G, which has a vertex of degree greater than two closest to the central edge (central vertex) by extending a radial path.

It remains to prove that nonbasic trees are reconstructible. Let G be a nonbasic tree, and let F be a basic tree with the same diameter as G. By the counting theorem, the number of maximal basic subtrees of G isomorphic to F is reconstructible. We can use this information to find the radial branches of G as follows. Each non-path radial branch which includes k peripheral vertices of G is the top of p(G) - k maximal basic subtrees of G, where p(G) is the number of peripheral vertices in G. This gives us the non-path radial branches of G (with multiplicities). The number of path radial branches is then p(G) minus the total number of peripheral vertices in the non-path radial branches.

In the central case, it still remains to reconstruct the nonradial branches. But they are just the nonradial branches of a G, obtained by deleting either a peripheral vertex of a radial branch which includes at least two peripheral vertices, if there is such a branch, or a nonperipheral end vertex of a radial branch, if there is such a vertex; otherwise, all radial branches are paths, and the nonradial branches can be found from a G, obtained by deleting a peripheral vertex I

After trees, it is, perhaps, natural to look at bipartite graphs in general. Unfortunately, there is nothing to report here besides the observation that bipartite graphs are recognizable. Bipartite graph reconstruction remains a challenging open problem.

The transition from trees to separable graphs, however, has proved more fruitful. For a separable graph G which is not a tree, we define the trunk of G to be the maximal subgraph of G that has no vertices of degree one. A limb of G is a maximal subtree that contains exactly one vertex of the trunk; this vertex is the root of the limb. By first reconstructing the blocks of a separable graph, Bondy (1969b) showed that both the trunk and the rooted limbs are reconstructible. He deduced that separable graphs are reconstructible in certain special cases-for example, when there are no vertices of degree one. But, in general, the problem of where to root the limbs in the trunk is unsolved. Greenwell and Hemminger (1969) showed how to do this when the “pruned center” (the center of the block-cutvertex tree of the trunk of G) has, as its automorphism


group, a subgroup of the dihedral group.* They thereby generalized earlier results on the reconstruction of unicyclic graphs (Manvel, 1969b) and cacti (Cieller and Manvel, 1969). Further extensions, obtained by refining the methods of Greenwell and Hemminger, have been given by Krishnamoorthy (1976) and Krishnamoorthy and Parthasarathy (1976b).

Two other tree-like classes of reconstructible graphs are the class of 2-trees (Le Fever and Ray-Chaudhuri, 1976) and the class in which all cycles pass through a common vertex (Manvel and Weinstein, 1976; see also O’Neil, 1974)-together with cacti, these include all graphs G such that E = Y + 1. Critical blocks have been reconstructed by Krishnamoorthy (1976).

For graphs of higher connectivity, there is a general theorem due to Greenwell and Hemminger (1973). It includes, as special cases, two results which we have already mentioned: the reconstruction of components and of blocks.

Theorem. n-connected subgraphs of G isomorphic to F is reconstructible.

If F is n-connected but G is not, the number of maximal

Proof. This, again, follows from the counting theorem. Here, we take as the class of n-connected graphs and Ce as the class of graphs which

are not n-connected. 3 is recognizable because the connectivity of a nontrivial connected graph G satisfies the identity

K ( G ) = ~ + min K(G,). I

As mentioned earlier, the reconstruction of parameters is intimately related to the reconstruction of classes: given a reconstructible parameter, each value of that parameter determines, in a natural way, a recognizable class of .graphs, namely those graphs on which the parameter takes that value.

Many basic parameters have now been reconstructed. Of these, un- doubtedly the most interesting is the dichromatic polynomial, reconstructed by Tutte (1976a) using results from an earlier paper (Tutte, 1967). The dichromatic polynomial of G is defined by the formula

u E V(G)

* Krishnamoorthy and Parthasarathy (1976b) point out a minor error in this paper; in the case rn = g = 3, there are counterexamples with three branches which need to be characterized and excluded from Theorem 3.10.


where the sum is over all spanning subgraphs F of G and o ( F ) denotes the number of components of F. One immediately obtains the chromatic polynomial

P(G, x)=(-l)”Q(G; -x, -1)

and, hence, the chromatic number.

ble parameters: In addition, the dichromatic polynomial yields three new reconstructi-

(1) the number of disconnected spanning subgraphs of G having a specified number of components in each isomorphism class;

(2) the number of connected separable spanning subgraphs of G having a specified number of blocks in each isomorphism class;

(3) the number of nonseparable spanning subgraphs with a given number of edges.

This result of Tutte nicely complements Kelly’s lemma, which provides information only on vertex-proper subgraphs. It also enables us to reconstruct the characteristic polynomial (see Clarke, 1972). For Pouzet (1976a) has related the characteristic polynomial to the number of Hamilton cycles by showing that if one of these parameters is reconstructible then so is the other-and the number of Hamilton cycles is reconstructible by (3). Pouzet’s proof rests on the following observations:

(i) for l s k s v , each closed walk of length k induces either a vertex-proper subgraph or a Hamilton cycle of G ;

(ii) by Kelly’s lemma, the number of such walks which induce vertex- proper subgraphs is reconstructible;

(iii) the total number of closed walks of length k (including Hamilton cycles when k = v) is given by the trace of Ak, where A is the adjacency matrix of G;

(iv) the trace of A k is A t + A k + . - +A:, where ( A 1 , A 2 , . . . , A , ) is the spectrum of G.

Tutte (1976b) also considers a generalization of the characteristic polynomial, the idiosyncratic polynomial. This is the characteristic polynomial of the matrix one obtains from the adjacency matrix on replacing each zero by an indeterminate p. Tutte shows that it, too, is reconstructible, and proves, by elegant matrix methods, that a graph is reconstructible if its idiosyncratic polynomial is prime. Godsil and McKay (1977) have extended this result as follows.


Theorem. Let G(x) denote the characteristic polynomial of G. Then G is reconstructible provided that G(-x - 1) and G‘(x) have no nontrivial common factor.

The method of proof of this theorem is based on ideas of H. S . Wilf. The theorem itself appears to be quite a strong one, even though Godsil and McKay note that its hypothesis is satisfied only by graphs whose vertex- deleted subgraphs are pairwise nonisomorphic.

One fundamental parameter which has yet to be reconstructed is the genus. A natural first step would be to prove that planar graphs are recognizable, but even this is unsettled. [Of relevance here is the characterization of “nearly planar” graphs by Wagner;* see O’Neil (1973)]. However, it has been determined that maximal outerplanar graphs (Man- vel, 1972) and, indeed, all outerplanar graphs (Giles, 1974b) are recon- s tructible.

We have already noted that regular graphs are reconstructible. More generally, if we define a vertex u of G to be bad when G has a vertex of degree d(v)- 1, it is easily seen that G is reconstructible provided that some vertex has no bad neighbors. Thus G is reconstructible if

where B is the set of bad extended this observation by

vertices of G. Nash-Williams (1974) has proving that G is reconstructible if

Dorfler (1972) calls vertices of a graph S-related if either they are equal or they are adjacent and their remaining neighbors are identical; the equivalence classes defined by this relation are the S-classes of the graph. Together with Imrich (Dorfler, 1972; Dorfler and Imrich, 1972), he proves that a graph G is reconstructible if, for some k k 2 , G has an S-class of cardinality k but none of cardinality k - 1 . (The hypothesis requires, in effect, that G have some S-class which is not “bad,” in the sense used above.) With the aid of this result, Dorfler (1975) shows that certain X-joins and lexicographic products are reconstructible, and that

K. Wagner, Fastplattbare Graphen. J. Combinatorial Theory 3 (1967) 326- 365. MR36#73.


some strong products are weakly reconstructible. In addition, he proves that all Cartesian products are reconstructible.

One can define another relation on the vertices of a graph as follows: vertices u and u of G are C-related if G-{x, y } = G - { u , u } implies { x , y} = { u , v}. Chinn (1971) shows that this relation is reconstructible and that a graph is reconstructible if it has a vertex which is C-related to every other vertex.

The theory which we have discussed so far has clearly been guided by the first principle of mathematical research: if you cannot solve a given problem, solve a purr of it (P6lya*). There is, of course, a second principle-be wise, generulize-of which reconstructors are equally aware. Here, the aim is to reconstruct a graph from some proper subset of its deck of cards.

Recall that a function defined on a class % of graphs is termed reconstructible if, for each graph G in 3, it takes the same value on all reconstructions of G. Let G be a graph and let f be a reconstructible function such that f(G)c_ V(G). An f-reconstruction of G is a graph H such that f ( H ) = f( G) and H, = G, for all u E f( G); f-reconstructible graphs and classes of graphs are defined in the obvious way.

The endvertex function, which assigns to each graph its set of endvertices, is clearly reconstructible, and several classes of graphs have been proved endvertex reconstructible. These include trees (Harary and Palmer, 1966) and various other types of separable graphs (Bondy, 1969b; Greenwell and Hemminger, 1969; Krishnamoorthy and Parth- asarathy, 1976b). A naive conjecture by Bondy (1969b), that all graphs with sufficiently many endvertices are endvertex reconstructible, was readily disproved by Bryant (1971). Krishnamoorthy and Parthasarathy (1976b) even constructed an infinite family of graphs which are not f-reconstructible for any reconstructible function f such that f(G) is disjoint from the pruned center of G.

Bondy (1969a) strengthened the endvertex reconstruction of trees (Harary and Palmer, 1966) by showing that trees are peripheral vertex reconstructible. This can be proved in a similar manner to our treatment of tree reconstruction, using the following Kelly-type lemma due to Bondy (1968): if f is a reconstructible function and each subgraph of G isomorphic to F contains the same number m of elements of f(G), where rn < n = If(G)l, then

s(F, G) = 1 s(F, G,Mn - m). u e f ( G )

G. Polya, How to solve it. Princeton University Press, Princeton, NJ (1945).

238 J O U R N A L OF GRAPH THEORY

Harary (1964) suggested another way to make reconstruction harder: reduce the number of cards by deleting from the deck all but one of each isomorphism type. More precisely, we call a graph H a set-reconstruction of G if, to each u E V(G) , there corresponds a w E V ( H ) such that H, = G, and, to each u E V(H), there corTesponds a w E V(G) such that G, = H,. Harary conjectured that all graphs with at least four vertices are set reconstructible. Manvel (1970a, 1970b, 1976) showed that many parameters and several classes (disconnected graphs, trees, separable graphs without endvertices) are set reconstructible; so, too, are unicyclic graphs (Arjomandi and Corneil, 1974) and outerplanar graphs (Giles, 1976a). Manvel (1970b), NeSetiil (1971), and Nebesky (1974) have reconstructed trees under still more stringent conditions. [Of interest in this connection is a general result of NeSetiil (1969) to the effect that many reconstruction theorems are valid for all trees provided that they are valid for all central trees].

One is led to ask how much information is really necessary to reconstruct a tree-or, indeed, any graph. Manvel (1969) proposes the following question: how many cards can be discarded at random from a full deck and still guarantee that the graph is reconstructible?*

2. EDGE RECONSTRUCTION

It is apparent from the foregoing discussion that, despite considerable effort, relatively little progress has been made towards a proof of the Reconstruction Conjecture. The classes of graphs which are known to be reconstructible are few in number and simple in structure, and include neither bipartite nor planar graphs-indeed, it has not even been estab- lished that planar graphs are recognizable. A related, and seemingly easier, conjecture was put forward by Harary (1964). Here, one is presented with a deck of cards bearing, not the vertex-deleted subgraphs of a given graph, but its edge-deleted subgraphs. As before, one is asked to determine the graph from this information.

To formulate the problem precisely, we define an edge reconstruction of a graph G to be a graph y such that

E ( H ) = E(G) and He = G, for all e E E(G).

* Stockmeyer (1976b) notes that there are two nonisomorphic graphs on six vertices havkg five vertex-deleted subgraphs in common. Thus the claim in Manvel (1 969a) that 5(6) = 5 is in error.

GRAPH RECONSTRU CTI ON-A SURVEY 239

FIGURE 4 Two non-edge-reconstructible pairs

[Note that E ( H ) = E(G) does not imply that V ( H ) = V(G).] Edge- reconstructible graphs and functions, and edge-recognizable and (weakly) edge-reconstructible classes of graphs, are defined in an entirely analog- ous manner.

As with vertex reconstruction, not all graphs are edge reconstructible; there are two small non-edge-reconstructible pairs (see Figure 4).

With these in mind, Harary formulated the

Edge Reconstruction Conjecture. All simple finite graphs with at least four edges are edge reconstructible.

In the remainder of this section we confine our attention to finite simple undirected graphs with at least four edges.

One’s intuitive feeling, that this conjecture is weaker than the Recon- struction Conjecture, is confirmed by a theorem of Harary and Palmer (1965): a graph is edge reconstructible if its edge graph (that is, line graph) is reconstructible. (In fact, as they conjectured and Hemminger (1969) proved, a graph is edge reconstructible if and only if its edge graph is reconstructible.)

The basic results that we shall need are

Kelly’s lemma (for edges). For any two graphs F and G such that E ( F ) < E ( G ) , the number of subgraphs of G isomorphic to F is edge reconstructible.

Counting Theorem (for Edges). Let 93 be an edge-recognizable class of graphs, and let 9 be any class of graphs such that, for every G in 9, each F-subgraph of G is

(i) edge proper; (ii) contained in a unique maximal $-subgraph of G.

Then, for every F in 9 and every G in 3, the number of maximal %subgraphs of G isomorphic to F is edge reconstructible.

The number of isolated vertices is readily shown to be edge reconstructible. By Kelly’s lemma, one can determine whether or not the longest path in G has length one, two, or more. The number of isolated vertices


of G is accordingly m - 2, m - 1 or m, where m denotes the minimum number of isolated vertices in any G,.

We can now prove that the Edge Reconstruction Conjecture is valid for all graphs if it is valid for graphs without isolated vertices. Let G be a graph with precisely n isolated vertices, and let H be an edge reconstruction of G. Because H also has exactly n isolated vertices, we can write G=G'+nK, and H=H'+nK, , where G' and H have no isolated vertices. Since He = G, for all e E E, Hi= G: for all e E E. If, now, G' is edge reconstructible, then G'=H' and so G=H. Hence G, too, is edge reconstructible.

Greenwell (1971) showed that, for a graph without isolated vertices, reconstructibility implies edge reconstructibility. Our proof makes use once more of the counting theorem,

Theorem. edge reconstructible.

Proof. For a graph G without isolated vertices, let 3 be the class of all edge reconstructions of G and let 9 be the class of graphs with Y - 1 vertices. Since edge reconstructions of G have no isolated vertices, their 9-subgraphs are edge proper and the counting theorem applies. But the maximal $-subgraphs of G are exactly the vertex-deleted subgraphs of G. It follows that G is edge reconstructible if G is reconstructible I

By combining this theorem with the results of the previous section, we immediately deduce that several classes of graphs are edge reconstructible; regular graphs, disconnected graphs with at least two nontrivial components, trees, certain products (see also Dorfler, 1974), and so on. Likewise, parameters such as the chromatic polynomial (see also Manvel, 1970a), the characteristic polynomial, the number of spanning trees and the number of Hamilton cycles are edge reconstructible.

However, much more can be said. Indeed, all graphs, except possibly those with relatively few edges, are edge reconstructible. This striking result, the proof of which is based on an ingenious application of the inclusion-exclusion principle by Lovisz (1972), is due to 'Muller (1977). We shall derive it from a theorem of Nash-Williams (1976).

First, a little notation. If G and If are graphs and F is a spanning subgraph of G, we let (G + H)F denote the set of injections 7i- : V(G) + V(H) such that, for each edge uv of G, T ( u ) ~ ( v ) is an edge of H if and

only if uv is an edge of F. For convenience, we denote the set of embeddings of G into H [that is, (G + H),] by G 4 H, and the number I(G+ W F ( by IG + HI,. We shall also write Fc G to indicate that F is a spanning subgraph of G.

If G is reconstructible and has no isolated vertices, then G is


Theorem. G is edge reconstructible if either of the following two conditions holds:

(i) there exists a spanning subgraph F of G such that IG + HI, =

(ii) there exists a spanning subgraph F of G such that E ( G ) - E ( F ) is

Proof. Let G and H be graphs. Then, for any G’G G

IG -+ GI, for every edge reconstruction H of G;

even and IG + GI, = 0.

Applying the Mobius inversion theorem,* we get

If H is, in fact, an edge reconstruction of G, it follows from Kelly’s lemma that

IG+ GIF-IG+HIF= 1 (-l)e(G’)-c(F)(lG’-+ Gl-/G’-+ HI) FSG’EG

= (-l)e(G)-E(F)(lG + GI-IG + HI).

Solving for IG + HI, we have

Since the first term on the right-hand side is positive, and the second is non-negative [whenever F and G satisfy condition (i) or condition ($1 IG -+ HI is positive. But this means that G and H are isomorphic, since they have the same number of edges I

Corollary 1. (LovAsz, 1972). G is edge reconstructible if E >;(:).

Proof. When F is the empty graph IG-,HIF=IG-,HCI. If E >$(;), this is zero for every edge reconstruction H of G, and so condition (i) of the theorem is satisfied I

* M. Hall, Jr., Combhatorial Theory. Blaisdell, Waltham, MA (1967), Theorem 2.2.1.

242 JOURNAL O f GRAPH THEORY

Schmeichel (1975) has strengthened Corollary 1 a little by giving a condition on the degree sequence of G which forces IG + WI to be zero for every edge reconstruction H of G (namely, di + d v + l - i 2 v for some i, where d , s d , 5 . . . 5 d v ) .

Corollary 2. (Miiller, 1977). G is edge reconstructible if 2'-'> v!

Proof. There are 2'-' spanning subgraphs F of G such that E ( G) - E ( F ) is even, and there are v ! permutations of V. So if 2"-' > v!, some F will satisfy condition (ii) of the theorem

A slightly weaker version of Corollary 2, valid for Y L 10, reads; G is edge reconstructible if E 1 Y log, (v/2). Although Corollary 2 is generally far stronger than Corollary 1, the latter gives a better bound when v < 10.

Note that the proof of the last theorem and that of the counting theorem have the same format: (1) produce a simple identity, (2) invert the identity, and (3) apply Kelly's lemma. Hopefully, this approach will lead to further substantial results.

I

3. DIGRAPHS

The question of reconstructibility applies equally well to directed graphs as to undirected graphs. Here, in addition to several other small counterexample pairs, one finds pairs of tournaments on three and four vertices (see Figure 5) .

In view of these counterexamples, it is natural to consider 'first the reconstruction of tournaments. This was the approach adopted by Harary and Palmer (1967). They proved that tournaments which are not diconnected (and have at least five vertices) are reconstructible and conjectured that the same is true of diconnected tournaments. We shall give a short proof of their theorem and then discuss the conjecture. Note that both Kelly's lemma and the counting theorem apply (with appropriate modifi- cations) to digraphs; in the counting theorem we shall denote the classes of digraphs by 9 and 9. Also recall that, in any tournament D, there is a natural linear ordering D,, D,, . . . , D, of its dicomponents such that, €or 1 5 i < j 5 n, each vertex of Di dominates every vertex of Di.

AA A A FIGURE 5 Nonreconstructible tournaments with three and four vertices.


Theorem. Nondiconnected tournaments on at least five vertices are reconstructible.

Proof. We first employ a theorem of Moon* to establish recognizability. Moon’s theorem states that, in a diconnected tournament D, each vertex is included in a directed cycle of every length k (3 I k I v). It follows easily that, for v 2 4, a tournament D is diconnected if and only if a t least two vertex-deleted subtournaments D, are diconnected. Since tournaments are clearly recognizable, so are diconnected tournaments.

Now let D be a nondiconnected tournament. To reconstruct D, it suffices to determine the dicomponents of D and their linear ordering. Since the dicomponents are vertex proper, they can be found by means of the counting theorem, taking 9 and 9 as the classes of diconnected and nondiconnected digraphs, respectively. It therefore remains to determine the linear ordering D,, D,, . . . , D, of the dicomponents of D. We consider two cases, depending on whether or not D has a receiver (a vertex which is dominated by every other vertex). Note that a tournament D which has a receiver contains at least v- 1 in-(v-2)-stars, whereas one with no receiver contains at most three in-(v-2)-stars. So, by Kelly’s lemma, tournaments with receivers and at least five vertices are recognizable.

If D has no receiver, choose a 0, with at least n dicomponents such that the total number of vertices in the first II - 1 dicomponents (in the linear ordering) is as large as .possible. Then these first n - 1 dicomponents are D,, D,, . . . , D,-l in that order. If D has a receiver, choose a 0, with exactly n- 1 dicomponents, in which the trivial dicomponents (if any) occur as early as possible in the linear ordering. These n - 1 dicomponents are then D,, D,, . . . , Since the remaining dicompo- nent D, is known, it follows that D is reconstructible I

When one turns to tournaments in general, one has the now familiar pattern of partial results. It is not difficult to show, for instance, that the score sequence (sequence of outdegrees) is reconstructible. This result, due To Harary and Palmer (1967), can also be proved by means of the counting theorem, taking for 9 the class of all outstars. (A slight complication arises if D has a transmitter and/or receiver, but these cases are recognizable and easily dealt with.)

In the case of tournaments, however, partial results are deceptive, for here the Reconstruction Conjecture is false. The original Harary-Palmer

J. W. Moon, On subtournaments of a tournament. Canad. Math. Bull. 9 (1966) 297-301. MR34 #95.


conjecture was disproved by Beineke and Parker (1970) when they discovered counterexample pairs on five and six vertices. A renewed conjecture by Harary (1974), excluding these counterexamples, was then refuted by Stockmeyer (1975), who found two pairs on eight vertices by means of an exhaustive computer search (which also determined that all tournaments on seven vertices are reconstructible). Finally, Stockmeyer (1976a, 1977) administered the coup de grice by constructing nonreconstructible tournaments on 2" +2" vertices, for all rn and n not both zero.

The basis of Stockmeyer's counterexamples is a remarkable family {A, I n>0} of tournaments. Let p =2". Then A,, has vertex set {ul, u2,. . . , up} and arc set {(u,, u,) I odd(j - i) = 1 (mod 4)) where, for any nonzero integer k, odd(k) is the odd integer obtained on dividing k by the appropriate power of 2. [Thus odd(-6)=-3 and odd@)= 1.3 As an example, the adjacency matrix of A, is displayed in Figure 6.

Each tournament A, possesses a fascinating combination of properties: it is self-converse, all of its subtournaments A, - ui are self-converse, the subtournaments A, - u, and A, - tjp+l.-i are isomorphic (1 5 i 5 p ) and, somewhat surprisingly in view of the above, its automorphism group is trivial.

By augmenting A, in two different ways, one obtains nonreconstructible tournaments B, and C,, on 2" + 1 vertices. In each case a new vertex uo is added, but, in B,, uo dominates u2, u4, . . . , up and is dominated by ul, u 3 , . * . , up-l, whereas, in C,, u,, dominates u l , u g , . . . , up-l and is dominated by u2, u4, . . . , up.

The adjacency matrices of tournaments B, and C, are shown in Figure 7.

It can be verified that B,& C,, but B, - uo= C, - uo and B, - u, = C, - (1 5 i 5 p). Curiously, B, and C, are also self-converse.

Stockmeyer (1976a) also notes that there are many nonreconstructible digraphs other than tournaments. One class is depicted in Figure 8.

0 0 0 1

1 0 0 0 1 0 1

h3: 0 1 0 0 0 1 1 0

0 0 1 0 0 0 1 1

I 1 0 0 1 0 0 0 1

1 1 1 0 0 1 0 0 0

FIGURE 6 The adjacency matrix of A3.


0

0

1

0

1

B 0

1

0

1

0

-

3:

0 1 0 1 a * . 0 1

0

0

1

0

1

0

1

0

1

- ) 1 0 1 0 1 0 1

c : A3 3

1 0 1 0 1 0 1 0

A3

FIGURE 7 The adjacency matrices of B3 and C,.

On the positive side Manvel (1973, 1974b, 1975) has proved that, for digraphs with at least five vertices, the degree-pair sequence is reconstructible and the connectedness type (diconnected, unilateral, connected, or disconnected) is recognizable. For the class of acyclic transitive digraphs, Das (1975b) strengthens the degree-pair sequence result of Man- vel by reconstructing the pairs (D-(u) , D'(v)), IJ E V, where D - ( v ) [respectively, D+(v) ] is the subdigraph induced by v and its in-neighbors (respectively, out-neighbors). Das (1973) has also reconstructed acyclic transitive digraphs which have a unique source. Finally, Harary and Palmer (1966) observe that their reconstruction of trees from the endvertex-deleted subtrees applies as well to oriented trees with at least three endvertices.

When we turn to arc reconstruction, the methods of the previous section can generally be applied. In particular, the inclusion-exclusion technique of L O V ~ S Z goes through unmodified, and one easily obtains as a corollary the result of Harary and Palmer (1967) that tournaments are arc reconstructible.

0

*"

FIGURE 8 A class of nonreconstructible digraphs other than tournaments.


4. INFINITE GRAPHS*

With infinite graphs, the reconstruction problem takes on an entirely new character. For here counterexamples abound, and one is forced to replace the conjecture: all infinite graphs are reconstructible by the question: which infinite graphs are reconstructible?

In retrospect, it is surprising that it took so long for counterexamples to emerge. The problem was proposed initially by Harary (1964) and again, for infinite trees, by Nash-Williams (1967). Fisher (1969) was the first to produce a nonreconstructible infinite graph. The simplest example, discovered independently by NeSetiil (1972) and Fisher, Graham, and Harary (1972), is T,, the regular tree of degree X,; it has 2T, as a reconstruction.

The pair (T-, 2Tm) demonstrates the impossibility of fully extending, to infinite graphs, Kelly’s reconstruction of trees and disconnected graphs; indeed, it shows that these classes are not even recognizable in the infinite case. Thus, in order to obtain positive results, some restriction must be imposed. A natural one is to limit attention to locally finite graphs (a class which is clearly recognizable).

However, this restriction is not, in itself, sufficient: Harary, Schwenk, and Scott (1972) have constructed an infinite family of nonreconstructible locally finite countable forests. It may be instructive to describe one of these, as it is typical in many ways of all known counterexamples. Let 3, denote the tree obtained by joining one vertex from each of n copies of T2 (the regular tree of degree two) to a new vertex. The graphs J,, J2, and J3 are depicted in Figure 9. Denote by 3 the class of graphs other than T’ which are almost 2-regular subtrees of J,, for some n. (A graph G is almost k-regular if all but a finite number of vertices of G have degree k ) . Let G be the disjoint union of KO copies of each tree in 3, and let H = G + T2.

J1 J2 FIGURE 9 The graphs J , , J p , and J3.

J3

* It is best not to carry a full set of cards (Harary, 1964).


Clearly, G and H are countable and G& H ; but H is a reconstruction of G since, for both G and If, the set of vertex-deleted subgraphs is (G + nT, 1 0 5 n < No} and the multiplicity of occurrence of each is X,.

Despite these counterexamples, some extensions of finite reconstruction results have been obtained. Bondy and Hemminger (1974) show that an infinite disconnected graph is reconstructible if either (a) all components are finite or (b) some finite component occurs only a finite number of times. Forests are treated in two papers. NeSetiil (1972) reconstructs forests that contain no one-way infinite paths, and Harary, Schwenk, and Scott (1972) reconstruct almost k-regular forests.*

Bondy and Hemminger (1974) have observed that locally finite trees are recognizable. Harary, Schwenk, and Scott (1972) conjecture that they are, in fact, reconstructible. We are inclined to agree with this, although the only locally finite trees which have been reconstructed so far (apart from the ones already mentioned) are (a) those which are rn-coherent. (exactly m distinct one-way infinite paths emanating from each vertex) for 1 s m <No (Bondy and Hemminger, 1974, for rn > 1; Thomassen, 1977, for m = l ) and (b) those with a finite nonempty set of endvertices (Lin, 1976). A similar class of trees, those which are almost (but not quite) locally finite, has been reconstructed by Krishnamoorthy and Parth- asarathy (1976~).

The distinguishing characteristic of all classes of infinite trees reconstructed to date is shared by finite trees-the members of the class have a finite “center” that is recognizable in “most” of the vertex-deleted subgraphs. To illustrate the techniques used, we outline a proof (which differs from the original one) that a locally finite tree G with a finite nonempty set S of endvertices is reconstructible. (The class of such graphs is clearly recognizable.) Define the center C of G to be the union of all paths (including those of length zero) between pairs of endvertices of G. Let T be the set of vertices of G which are not in C and are not joined to any vertices of degree one or two. One easily recognizes a G, with u in T. In any such Guy exactly one component has endvertices, and the center of this component is precisely the center of G. The proof divides into two cases, depending on whether T is a finite or an infinite set; we only consider the latter case here. For u in T, let H, be the component of G, which contains C and let Hf: be the subgraph of H,, consisting of C and all paths of length at most k which start in C. Denote the (unique) maximal element of {Hf:}uET (obtained when u is any vertex of T at a

*According to Thomassen (19771, t h e assertion on page 504, line 11 of NeSetiil (1972). that T,,=0, is in error.


distance greater than k from C) by Hk. Then

G = lim Hk k -m

(See Bondy and Hemminger, 1974, for details of this limiting procedure.) The above approach seems to be suitable only for very special classes

of locally finite trees; it is not likely to lead to a proof of the Harary- Schwenk-Scott conjecture.

Turning to edge reconstruction of infinite graphs, we again come up against the seemingly inevitable counterexamples. By means of a rather complicated construction, Thomassen (1976) has shown that, for every infinite cardinal a, there exists a graph with IY edges which is not edge reconstructible. As he points out, the edge graphs of these graphs furnish additional examples of nonreconstructible infinite graphs (since an infinite graph is edge reconstructible if its edge graph is reconstructible). For the same reason, most reconstruction results for disconnected infinite graphs still hold in the case of edge reconstruction. However, it is not known whether the reconstructibility of an infinite graph implies its edge reconstructibility, since the counting theorem obviously dces not apply. Simi- larly, we lose the Mobius inversion technique employed so effectively in the finite case.

Harary's extension of the Reconstruction Conjecture to infinite graphs has given rise to many counterexamples, and these in turn have forced us to concentrate on a very restrictive, albeit interesting, class of infinite graphs, namely, locally finite trees. Fortunately, there is another, perhaps more natural, extension due to R. Halin (see Bondy and Hemminger, 1975). Halin observed that all known nonreconstructible pairs have the property that each is isomorphic to an induced subgraph of the other. We call graphs with this property 2-isomorphic," and say that a graph G is %-reconstructible if every reconstruction of G is %-isomorphic to G. The definitions of %'-reconstructible and weakly %-reconstructible classes follow the earlier pattern.

Halin's conjecture. All graphs are %-reconstructible.

This conjecture merits attention since it not only generalizes the Reconstruction Conjecture, but also excludes all existing counterexamples to the conjecture proposed originally by Harary. All that has been

* In the theory of relations, the notion of %?-isomorphism was introduced by R . Fraisse, Sur la comparaison des types de relations. C. R. Acad. Sci. Paris Ser. A 226 (1948) 987-988. MR 10 p. 517.


proved so far is that disconnected graphs are weakly %'-reconstructible (Bondy and Hemminger, 1975).

5. RELATED QUESTIONS

Nearly half of the entries in the bibliography deal either with the reconstruction of structures other than graphs (a research topic first proposed by Ulam, 1960) or with the reconstruction of graphs from other information. We shall review the literature briefly; readers interested in a particular topic should consult the pertinent papers for details.

Reconstruction of Structures other than Graphs.

1. Colored Graphs. A colored graph is a pair (G, 4), where G is a graph and 4 is a coloring (not necessarily proper) of its vertices and edges. The graph (G, 4) is vertex-colored (respectively, edge-colored) if 4 is constant on edges (respectively, vertices). Two colored graphs (G, 4) and (H, +) are isomorphic if there is an isomorphism between G and H such that corresponding vertices and edges are colored alike. Manvel (1970a) conjectures that all colored graphs with at least three vertices are reconstructible. The conjecture has been verified for colored graphs with at most five vertices (Weinstein, 1975), and for vertex-colored graphs with at most seven vertices (Manvel, 1970a).

In a colored graph, one can think of the color distribution of loops at a vertex u, together with the color of u, as a new color assigned to vertex u. Similarly, the color distribution of edges joining two vertices u and v can be regarded as a new color assigned to edge uu. When discussing colored graphs, therefore, it suffices to consider those that are simple. (Note, also, that colored simple graphs are equivalent, from the point of view of recqnstruction, to uncolored graphs with loops and multiple edges). Pouzet (1977) has shown that the problem can be further reduced to edge-colored simple graphs, thereby extending an earlier result due to Weinstein (1975).

Weinstein (1975) calls a colored graph (G, 4) refinably reconstructible if (G, 4) is reconstructible for all refinements J, of 4. By analyzing this concept, he is able to deduce results such as the following: a colored graph is reconstructible if each vertex color class contains at most three vertices. He also points out that many reconstruction results carry over, with little or no difficulty, to colored graphs. In particular, both Kelly's lemma and the counting theorem hold; and colored disconnected graphs,


colored trees, and colored separable graphs without endvertices are reconstructible.

The reconstruction of a class of graphs can sometimes be based on the reconstruction of a class of colored graphs. For instance, if G has a vertex u that can be identified in G,, for each u f . u and if every 2-vertex- coloring of G, is reconstructible, then G itself is reconstructible. This idea is employed by Manvel and Weinstein (1977) in the reconstruction of nearly acyclic graphs (graphs in which some vertex-deleted subgraph is a forest). Similarly, Giles (1974b) reconstructs outerplanar graphs by fist reconstructing certain colored outerplanar graphs.

The reconstruction of bipartite graphs was cited earlier as a worthwhile goal. A little easier (perhaps) would be the reconstruction of properly 2- vertex-colored bipartite graphs. Some observations on this problem are made by Hyrro (1968).

A vertex-colored graph in which a t most one color class contains more than one vertex is said to be partially labelled. Harary and Manvel (1970) consider partially labelled graphs on p vertices in which the exceptional color class contains n vertices. They define r(p, n ) to be one more than the maximum number of vertex-deleted subgraphs that two nonisomcr- phic graphs of this type can share. Their results include the following:

r (p , n ) r [ ( n + 1)/2]+2, 1 s n p - 1

r (p , 3 ) = r ( p , 4) = 4.

They conjecture that r (p , n ) = n for 4 I n 5 p, and verify this for p 5 7

2. Hypergraphs. By a hypergraph G, we mean a nonempty family (Ei I i E M) of nonempty sets. The elements of U iEMEi are the vertices of G, and the sets Ei are the edges of G. We write V ( G ) = U i e M Ei and E ( G ) = (E, I i EM). For S c V(G) , the hypergraph G[S] = (Ei I Ei c_ S, i E M) is the section hypergraph induced by S. For I c M, the hypergraph GII]=(Ei I i d ) is the partial hypergraph induced by I. Hypergraphs G = (Ei I i E M) and H = (F, I i EM) are isomorphic if there is a bijection C$ : V ( G ) 4 V ( H ) and a permutation 7~ : M + M such that (b(Ei) = F& for all i E M ; G and H are strongly isomorphic if 7~ can be chosen as the identity. A reconstruction result involving strong isomorphisms in the hypothesis and conclusion will be referred to as a strong reconstruction.

Faber (1974, 1975), states four families of conjectures on hypergraphs


of rank r (those in which no edge has more than r vertices). One of these includes the Reconstruction Conjecture, and another the Edge Recon- struction Conjecture. Faber conjectures, for example, that two hypergraphs G and H of rank r on the same vertex set V are isomorphic if, for some k 2 r and for every k-element subset S of V, their section hypergraphs G[S] and H [ S ] are isomorphic. He proves this conjecture for k 5 v - r, and shows that its truth for k = v - 1 would imply its truth for all smaller values of k.* Daber's result was proved originally for graphs (the case r = 2 ) by Kelly (1964).

In treating infinite hypergraphs, Berge and Rado (1972) and Rado (1973, 1974):

(1) Show that G is strongly reconstructible from its partial hypergraphs induced by finite index sets, provided that all the edges of G are finite;

(2) characterize the hypergraphs G, with M and V finite, that are not strongly reconstructible from their partial hypergraphs induced by index sets of a fixed size p<IMI, in particular, from their edge- deleted partial hypergraphs; and

(3) construct, for each nonempty set M, a hypergraph G, with edges indexed by My that is not strongly reconstructible from its edge- deleted partial hypergraphs.

By statement (1) we mean: if G = ( E i l i E M ) and H = ( F , ( i E M ) are hypergraphs with the property that the partial hypergraphs (Ei I i E I ) and (F, 1 i E I) are strongly isomorphic for every finite subset I of M, then G and H are strongly isomorphic. The meanings of (2) and (3) should now be clear.

Rado and Wiikie (1976) strengthen (1) above, as follows. Consider the class of hypergraphs of rank at most n, and let f(n) denote the smallest integer rn such that all graphs in this class are strongly reconstructible from their partial hypergraphs induced by index sets of size rn. Then f(n) = 2 +[log, n] for n 2 1.

Readers interested in pursuing this aspect of reconstruction are referred to the book by Berge?

3. Relations. An n-ary relation on a nonempty set V is a function G : V" + { +, -}, where V" is the set of ordered n-tuples on V. (Simple graphs are, of course, symmetric irreflexive binary relations.) If S c V, we denote by G I S the relation which results on restricting the relation G to S".

*The conjecture has now been disproved by 0. J. A. Welsh (personal communication).

C. Berge. Graphs and Hypergraphs. North-Holland, Amsterdam (1 973).


Most of the work in this area stems from Frasnay’s (1965) reconstruction of a special class of relations called enchainable relations and is in response to a conjecture of FraissC (1971): given any positive integer n, there exists an integer f(n) such that if G and H are n-ary relations on a set V of at least f(n) elements, and if G I S and H 1 S are isomorphic for every proper subset S of V, then G and H are isomorphic.

Various extensions of Frasnay’s result are given by Pouzet (1972), FraissC and Pouzet (1971), and Lopez (1975). Lopez (1972a, b) proves FrakC’s conjecture for all binary relations and, more generally, for all binary multirelations (1977). Pouzet (197.5) has shown, however, that for n 2 3 the conjecture is false.

4. Matroids. Brylawski (1974) discusses a number of reconstruction conjectures for matroids, including natural analogs of the Reconstruction Conjecture-a binary matroid is reconstructible from its collection of restrictions to hyperplanes-and the Edge Reconstruction Conjecture-a matroid other than a circuit or Boolean algebra is reconstructible from its element-deleted submatroids.

An example is given to show that the specialization to binary matroids in the former conjecture is essential. However, even €or binary matroids, it appears to be difficult to reconstruct almost anything (including the number of elements). Counterexamples to the latter conjecture are given in Brylawski (1975).

5. Miscellaneous. Other activity includes the reconstruction of square- celled animals (Harary and Manvel, 1972), the reconstruction of matrices (Manvel and Stockmeyer, 1971), the reconstruction of the characteristic polynomial of a graph from the characteristic polynomials of its vertex- deleted subgraphs (CvetkoviC and Gutman, 1975), the weak edge reconstruction of the automorphism group for certain classes of graphs (Sheehan, 1973), an abstract concerning a generalization of the Edge Reconstruction Conjecture (NeSetiil, 1973), and a series of papers about the reconstruction of To-topologies (Das, 1973, 197Sa, b). The latter papers were discussed in Sec. 3, since there is a natural correspondence between To-topologies and acyclic transitive digraphs.

Reconstruction of Graphs from Other Information

1. k-Vertex-Deleted Subgraphs. The idea of generalizing the Recon- struction Conjecture to k-vertex-deleted subgraphs with k > 1 was first mentioned by Kelly (1957), who observed that not all graphs are determined by their 2-vertex-deleted subgraphs. It appears, however, that all

GRAPH RECOFISTRUCTION-A SURVEY 253

sufficiently large graphs are so determined, and Manvel (1969a) has proposed the following conjecture: given any posiliue integer k, there exists an integer f (k) such that any graph on at least f(k) vertices is reconstructible from its collection of k-uertex-deleted subgraphs.

Nydl (1976) constructs examples which show that f( k ) L 2 k - I (if such a function indeed exists), thereby extending an earlier lower bound due to Manvel (1974a). On the other hand, Miiller (1976) proves that, for every c>O, almost all graphs on n vertices are reconstructible from their (n /2) ( 1 - c)-vertex-deleted subgraphs.

The conjecture has been verified by Giles (1976b, c) for trees (see also Spencer, 1969) and, in the case k = 2, by Manvel (1974a) for disconnected graphs (except when there are exactly two components, one of which is isomorphic to K , ) .

2 . Elementary Contractions. Bhave and Sampathkumar (1976) show that, with minor exceptions, disconnected graphs are reconstructible from their elementary contractions (obtained by identifying pairs of adjacent vertices), from their elementary homomorphic images (obtained by identifying pairs of nonadjacent vertices), and from their elementary partitions (obtained by identifying pairs of vertices). Bhave, Kundu, and Sampath- kumar (1976) likewise reconstruct trees from this information, while Kundu (1974) reconstructs unicyclic graphs from their elementary contractions. Also using elementary contractions, Krishnamoorthy and Parth- asarathy (1976a) reconstruct various classes of separable graphs. Related reconstruction results are given by Devadas, Jayaram, and Sampathkumar (1977) and Devadas and Sampathkumar (1977a, b, c).

3 . Spanning Trees. SedlAEek (1974, 1975) considers the reconstruction of graphs from their spanning trees. He finds that the wheels and the complete bipartite graphs K2." are so reconstructible, but that K2,+, is not. He also gives an example to show that nonisomorphic finite graphs can have the same collection of spanning trees (see Figure 10). Boyle

FIGURE 10 Nonisomorphic finite graphs with the same spanning trees.


(1976) has proved that K,,,,, and all subgraphs of K,,,, are reconstructible from their spanning trees.

4. Miscellaneous. Smolenskii (1962) proves that a tree can be reconstructed from the matrix of distances between its endvertices. Bondy (1969~) examines the reconstruction of graphs from their ‘closure functions’.

Legitimate Decks

One question which appears to be as fundamental as the reconstruction problem itself is the so-called legitimate deck problem (Harary, 1969). A deck is a coIlection of n graphs each having n - 1 vertices. A deck (Hi 115 is n ) is legitimate if there is a graph G with vertex set { 1,2, . . . , n} such that Gi =Hi for 1 4 i 5 n. The legirirnate deck problem asks for a characterization of legitimate decks. Very little has been published on this problem. What there is can be found in Greenwell and Hemminger (1969), O’Neil (1970), Smadici and Smadici (1972), Simpson (1974), RandiC (1976) and Rarnachandran (1977).

6. UNSOLVED PROBLEMS

Our choice of problems for this section has inevitably been guided by existing results. In particular, we have made an effort to select several which involve the “next step” in an approach to the general problem. For example, our first problem can be viewed as the next step following the reconstruction of regular graphs. It was suggested to one of the authors by R. W. Robinson in 1969.

Problem 1. Show that bi-degreed graphs are reconstructible. (This is trivial unless the two degrees differ by one.)

The next two problems have already been mentioned in the text.

Problem 2. Show that bipartite graphs are reconstructible.

Problem 3. Show that separable graphs with endvertices are reconstructible. (Even the reconstruction of separable graphs with two blocks, one of which is K2, would be a worthwhile achievement.)

A solution to Problem 3, combined with Bondy’s reconstruction of separable graphs without endvertices, would leave us with nonseparable graphs to reconstruct. Note, however, that solving Problem 3 under the


assumption that nonseparable graphs are reconstructible would also achieve this end and might well be an easier task. Our next problem expresses this philosophy in a general setting.

Problem 4. Let % be a recognizable class of graphs. Show that all graphs are reconstructible if % is reconstructible.

The genus is one of the few major parameters which have not yet been reconstructed; another is the edge chromatic number.

Problem 5. Show that the genus and the edge chromatic number are reconstructible.

Recall that a reconstructible parameter determines, in a natural way, a recognizable class of graphs. Showing that a class is recognizable, however, is only part of the job; the other part is the subject of our next problem.

Problem 6. Let f be any parameter (preferably an interesting one). Show that the class of graphs on which f takes a given value is weakly reconstructible.

From among the classes defined in this manner, we single out one for special mention.

Problem 7. Show that planar graphs are reconstructible. (One might first try to show that maximal planar graphs are reconstructible.)

Nebeskq (1976) has shown that, if G is a graph and if H is a subdivision of G with E ( I I ) Z ~ ( E ( G ) ) * , then H is reconstructible. This suggests the following problem.

Problem 8. Prove that G is reconstructible provided that E 5 v + f( v), where f(v) + ~0 as v + w.

The above problems remain open when we turn to edge reconstruction (except for the edge reconstructibility of the edge chromatic number). Indeed, since a graph without isolated vertices is edge reconstructible if it is reconstructible, it is advisable to attempt the edge versions of those problems first. Rather than state the edge analogs of all the previous problems, we pick out two of particular interest.

Problem 9. Show that bipartite graphs are edge reconstructible.

Problem 10. Show that planar graphs are edge reconstructible. Note that graphs of genus zero are edge recognizable via Kuratowski’s

theorem. The edge reconstruction of the genus in general, however, has not been considered.


Our next question concerns a possible coonection between planar duals.

Problem 11. Let G and H be planar duals. Is the edge reconstruction of G equivalent to the contraction reconstruction of H? (Some connectivity assumptions might be necessary.)

The Lavslsz and Miiller theorems were major advances, but they differ from most other reconstruction results in that their proofs are existential in nature. Constructive proofs might well be enlightening.

Problem 12. Give constructive proofs of the Lovslsz and Muller theorems.

Although the Digraph Reconstruction Conjecture has been disproved, we could, of course, ask about the reconstructibility of various classes of digraphs. We refrain, and instead state two problems of Stockmeyer ( 1 977a) concerning his counterexamples.

Problem 13. Are B, and C,, the only odd-ordered nonreconstructible tournaments?

Problem 14. Are all even-ordered counterexample pairs of tournaments complements of one another?

For infinite graphs, as for digraphs, it seems more appropriate to pose problems in the form of questions. The general question is “which infinite graphs are reconstructible?” The proposal of Nash-Williams (1967), to study infinite trees, is still of considerable interest, even though this class is not recognizable.

Problem 15. Are infinite trees weakly reconstructible?

While forests (even locally finite ones) are not reconstructible, the extension of the last problem to forests with a fixed finite number of components is also open. Indeed, a still more general question remains unsettled. Problem 16. Are infinite graphs with a fixed finite number of components weakly reconstructible?

The fact that the class of infinite connected graphs is not recognizable suggests the following problem.

Problem 17. Find necessary and sufficient conditions on the G,, G an infinite graph, for G to have a disconnected reconstruction.

Specializing Problem 15, we obtain one of the most important open problems concerning infinite graph reconstruction. It was first posed by


Harary, Schwenk, and Scott (1972), who conjectured an affirmative answer.

Problem 18. Are locally finite trees reconstructible?

The longer the list of nonreconstructible infinite graphs grows, the more Halin’s conjecture appears to be the proper extension of the Reconstruction Conjecture to infinite graphs. It warrants serious scrutiny.

Problem 19. Are all infinite graphs %‘-reconstructible?

For each of the above classes, edge reconstructibility is also an open problem. In fact, very little is known about the edge reconstruction of infinite graphs. An obvious, and desirable, remedy for this situation would be an extension of Greenwell’s theorem to infinite graphs.

Problem 20. Is an infinite graph edge reconstructible if it is reconstructible?

We feel that one other problem on the edge reconstruction of infinite graphs is of prime importance.

Problem 21. Are locally finite graphs edge reconstructible?

Clearly, many problems dealing with the material surveyed in Sec. 5 could be listed. We limit ourselves to four, since our main concern has been with the reconstruction and edge reconstruction of simple graphs, digraphs, and infinite graphs. The first assumes the validity of the Recon- struction Conjecture:

Problem 22. Show that all finite graphs are reconstructible if all simple finite graphs are reconstructible.

Our next problem also concerns colored graphs. It is a conjecture due to D. Kleitman (see Harary and Manvel, 1970).

Problem 23. Show that

where r (p , n ) is the function defined in Sec. 5.

An interesting question on binary relations has been suggested by Pouzet (1976b).

Problem 24. Find the smallest positive integer k for which the following assertion holds: there exists an integer n such that if G and H are binary


relations on a set V of v r n elements and if G 1 S and H I S are isomorphic for every subset S of V with 1st = Y- k, then G and H are isomorphic.

The digraph counterexamples of Stockmeyer (1976a) and the proof of Frakk’s conjecture for binary relations by Lopez (1972a, b) imply that 2 5 - 5 6 .

Problem 25. Find a (nontrivial) characterization of legitimate decks.

legitimate suggest themselves:

We close with the legitimate deck problem.

Several natural necessary conditions for a deck (Hi I 1 s i 5 n) to be

(a) For every graph F with v (F) < n

where s’(F,Hi) denotes the number of induced subgraphs of H, isomorphic to F.

(b) The “symmetric array” condition (Randit, 1976; Simpson 1976): the vertex-deleted subgraphs of the cards Hi can be arranged in a symmetric n x n array so that, for 1 5 i S n , the vertex-deleted subgraphs of Hi appear as the nondiagonal entries of row i.

(c) The degree sequences of the cards are compatible (in an obvious sense). W. Jackson of the University of Waterloo has constructed infinite families of illegitimate decks which satisfy both (a) and (b), both (b) and (c), and both (a) (i) and (c). Examples have also been constructed by U. Hafstram of Odense University.

Acknowledgments

We wish to thank several people for their helpful comments on the preliminary version of this paper, in particular Lowell Beineke, Willibald Dorfler, Stanley Fiorini, Frank Harary, Paul Kelly, Ben Manvel, Maurice Pouzet, Allen Schwenk, Paul Stockmeyer, Carsten Thomassen, Bjarne Toft, and Joe Weinstein. We are also grateful for the facilities provided by the University of Reading, England, where much .of the work was done.

The research of the first author was supported in part by a grant from the National Research Council of Canada.


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Graph reconstruction—a survey