An algebraic formulation of the graph reconstruction conjecture
The graph reconstruction conjecture asserts that every finite simple graph on at least three vertices can be reconstructed up to isomorphism from its deck - the collection of its vertex-deleted subgraphs. Kocay’s Lemma is an important tool in graph reconstruction. Roughly speaking, given the deck of a graph and any finite sequence of graphs, it gives a linear constraint that every reconstruction of must satisfy.
Let be the number of distinct (mutually non-isomorphic) graphs on vertices, and let be the number of distinct decks that can be constructed from these graphs. Then the difference measures how many graphs cannot be reconstructed from their decks. In particular, the graph reconstruction conjecture is true for -vertex graphs if and only if .
We give a framework based on Kocay’s lemma to study this discrepancy. We prove that if is a matrix of covering numbers of graphs by sequences of graphs, then . In particular, all -vertex graphs are reconstructible if one such matrix has rank . To complement this result, we prove that it is possible to choose a family of sequences of graphs such that the corresponding matrix of covering numbers satisfies .
The graph reconstruction conjecture was proposed by Ulam  and Kelly . Informally, it states that if two finite, undirected, simple graphs on at least three vertices have the same collection (multi-set or deck) of unlabelled vertex-deleted subgraphs, then the graphs are isomorphic; in other words, any such graph can be reconstructed up to isomorphism from the collection of its unlabelled vertex-deleted subgraphs.
The conjecture has been verified by McKay  for all undirected, finite, simple graphs on eleven or fewer vertices. In addition, it has been proven for many particular classes of graphs, such as regular graphs, disconnected graphs and trees (Kelly ). In fact, Bollobás  showed that for almost all graphs, just three (carefully chosen) subgraphs in the deck are sufficient to reconstruct the graph. On the other hand, a similar conjecture does not hold for directed graphs: Stockmeyer [11, 12] constructed a number of infinite families of non-reconstructible directed graphs. For a more comprehensive introduction to the problem, we refer to a survey by Bondy . For the standard graph theoretic terminology not defined here, we refer to West .
Kelly’s Lemma  is one of the most useful results in graph reconstruction. Let denote the number of subgraphs of isomorphic to . Kelly’s lemma states that for , the parameter is reconstructible, in the sense that if and have the same deck then . Several propositions in graph reconstruction rely on this useful lemma.
Kocay’s Lemma  allows us, to some extent, to overcome the restriction in Kelly’s lemma. It provides a linear constraint on that must be satisfied by every reconstruction of . Informally, it says that, if is a sequence of graphs, each of which has at most vertices, then there are constants such that the value of the sum is reconstructible, where the sum is taken over all unlabelled -vertex graphs . Roughly speaking, the constant counts the number of ways to cover the graph by graphs in the sequence .
Kocay’s Lemma has been used to show several interesting results in graph reconstruction. For instance, by carefully selecting the sequence , it is possible to give a simple proof that disconnected graphs are reconstructible. In addition, it can be used to show that the number of perfect matchings, the number of spanning trees, the characteristic polynomial, the chromatic polynomial, and many other parameters of interest are reconstructible; see Bondy .
It is natural to wonder whether even more restrictions may be imposed on the reconstructions of by applications of Kocay’s Lemma. Recall that it is possible to use different sequences of graphs in each invocation of the lemma, and as explained before, for each sequence we get a linear constraint that the reconstructions of must satisfy. By analysing such equations one would expect to obtain a wealth of information about the structure of any reconstruction of (perhaps enough equations may even allow us to conclude that is reconstructible). In this paper we investigate how much information one can obtain by setting up such equations.
We prove that the equations obtained by applying Kocay’s Lemma to the deck of a graph using distinct sequences of graphs provide important information not only about the reconstructions of , but also on the total number of non-reconstructible graphs on vertices. More formally, let be the number of distinct decks obtained from -vertex graphs. We show that if is the matrix of coefficients corresponding to these equations, then , i.e., the rank of this matrix provides a lower bound on the number of distinct decks. In particular, the existence of a full-rank matrix of coefficients would imply that all graphs on vertices are reconstructible. In addition, we give a proof that there exist sequences of graphs , with corresponding matrix of covering numbers, such that . In other words, if the graph reconstruction conjecture holds for graphs with vertices, then there is a corresponding full-rank matrix certifying this statement.
We state our results in more generality for graphs, hypergraphs, directed graphs, and also for classes of graphs for which similar equations can be constructed; for example, analogous results hold for planar graphs, disconnected graphs and trees.
Similar system of equations where considered by Kocay , where he restricted the total number of edges appearing in each sequence of graphs on a given system of equations to be the same. Interestingly, in this case it is not possible to show an equivalence to the graph reconstruction conjecture. In particular, Kocay computed the ratio of the number of independent edge-identities and the number of mutually non-isomorphic graphs with vertices, edges, and no isolated vertices (for small parameters ), and observed that these values can be strictly less than one. Kocay asked if the reconstruction conjecture would fail to be true if the ratio became small enough.
Our contribution may be summarised as follows. We remove the restriction, as in Kocay’s paper [Theorem 5.2 in 6], that the total number of edges be fixed among the sequences of graphs used to derive edge-identities. We show that the number of independent equations available at our disposal is precisely the number of distinct decks on a given number of vertices. Thus we give an algebraic characterisation, based on Kocay’s lemma, for the discrepancy between the number of different decks and the number of distinct graphs - a measure of how badly Ulam’s conjecture would fail to hold, if indeed it were to be false. In view of the result of Bollobás mentioned earlier, the ratio of the number of independent equations and the number of distinct graphs cannot be small.
A different mathematical perspective on such equations is presented in Mnukhin , where reconstruction problems are discussed in the more general context of orbit algebras. Mnukhin’s paper also mentions a formulation of Ulam’s conjecture in algebraic terms, based on whether the graph algebra is generated by disconnected graphs only. While there may be a translation between the two formulations, this is not immediately obvious to the authors. We refer the reader to Mnukhin’s paper for further details, and to the original reference  (in Russian) discussed in . Our results are simple to prove, can be specialised to several classes of graphs, as well as generalised to digraphs and hypergraphs, and provide an exact characterisation of the maximum number of independent equations.
Finally, our results may also be viewed as a limitation of the lemmas of Kelly and Kocay (which is proved using Kelly’s lemma), and in this regard we share the pessimism expressed by Tutte (see Chapter 9, page 113, ). The fact that the number of independent equations is equal to the number of decks suggests that the difficulties with Ulam’s conjecture lie somewhere else. In particular, it seems unlikely that applications of Kelly’s lemma and Kocay’s lemma will shed light on these difficulties.
In this paper, we consider general finite graphs - undirected graphs, directed graphs, hypergraphs, graphs with or without multiple edges, and with or without loops. We take the vertex set of a graph to be a finite subset of . We write for the family of -element subsets of a set . Further, we use the notation and .
Definition 2.1 (Graphs).
A hypergraph is a triple , where is its
vertex set (also called ground set, and written as
) and is its set of hyperedges (written as ),
and a map . An undirected graph is a
hypergraph with the restriction that ; in this case we call a hyperedge an edge (if
) or a loop (if ). An
undirected graph is simple if it contains no loop. A
directed graph is a triple , where is its
vertex set and is the set of its arcs, and a map . The first element of is called the
tail of the arc , and the second element of is
called the head of . We denote the set of all finite graphs
(including hypergraphs, undirected graphs and directed graphs) by
Although our results and proofs are stated in full generality, it may be helpful in a first reading to consider only finite, simple, undirected graphs.
Definition 2.3 (Graph isomorphism).
Let and be two graphs. We say that and are isomorphic (written as ) if there are one-one maps and such that an edge and a vertex are incident in if and only the edge and the vertex are incident in . Additionally, in the case of directed graphs, a vertex is the head (or the tail) of an arc if and only if is the head (or, respectively, the tail) of . The isomorphism class of a graph , denoted by , is the set of graphs isomorphic to .
A class of graphs is a set of graphs that is closed under isomorphism. A class of graphs is said to be finite if contains finitely many isomorphism classes.
Definition 2.5 (Reconstruction).
Let be graph and let be a vertex of . The induced subgraph of obtained by deleting and all edges incident with is called a vertex-deleted subgraph of , and is written as . We say that is a reconstruction of (written as ) if there is a one-one map such that for all , the graphs and are isomorphic. The relation is an equivalence relation. We say that a graph is reconstructible if every reconstruction of is isomorphic to (i.e., if implies ). A parameter is said to be reconstructible if for all reconstructions of . Let be a class of graphs. We say that is recognisable if, for any , every reconstruction of is in . Furthermore, we say that is reconstructible if every graph is reconstructible.
Let be a hypergraph. The number of edges incident with all vertices (i.e., edges such that , which we call big edges), is not a reconstructible parameter. For example, if is a graph obtained from by adding new edges and making them incident with all vertices in , then is a reconstruction of . In this sense, no hypergraphs are reconstructible, and each hypergraph has infinitely many mutually non-isomorphic reconstructions. If is a graph in class , then is not recognisable if for some , the graph is not in ; and is not finite if graphs are all in . On the other hand, the number of small edges, i.e., edges such that , is a reconstructible parameter.
In view of the above example, we will always use for the set of all graphs, for the set of all graphs without big edges, and for the set of -vertex graphs without big edges. A class will always be a subset of . We will use the following slightly restrictive definitions for some other reconstruction terms.
A graph in is reconstructible if it is reconstructible modulo big edges, i.e., if is a reconstruction of and , then is isomorphic to . A subclass of is recognisable if for each graph in , each reconstruction of in is also in . A subclass of is reconstructible if each graph in is reconstructible (modulo big edges).
Disconnected undirected graphs on 3 or more vertices are recognisable and reconstructible. However, there are classes of graphs that are recognisable, but not known to be reconstructible. An important example is the class of planar graphs (Bilinski et al. ).
Since and are equivalence relations, the quotient notation may be conveniently used to define various equivalence classes of graphs. We write the set of all isomorphism classes of graphs as ; analogously we use , , , and so on. We define an unlabelled graph to be an isomorphism class of graphs. But sometimes we abuse the notation slightly, e.g., if a quantity is invariant over an isomorphism class , then in the same context we may also use to mean a representative graph in the class. Similarly, we denote various reconstruction classes by , , , , and so on. Note that equivalence classes of any class of graphs under are refined by ; in particular, , and equality holds if and only if the class is reconstructible. We will refer to reconstruction classes of (i.e., members of ) by , and isomorphism classes of (i.e., members of ) by .
Given graphs and , the number of subgraphs of isomorphic to is denoted by . The following two subgraph counting lemmas are important results about the reconstructibility of the parameter .
Lemma 2.9 (Kelly’s Lemma, ).
Let be a reconstruction of . If is any graph such that , then .
Let be a graph and let be a sequence of graphs. A cover of by is a sequence of subgraphs of such that , , and . The number of covers of by is denoted by .
Lemma 2.11 (Kocay’s Lemma, ).
Let be a graph on vertices. For any sequence of graphs , where , , the parameter
is reconstructible, where the sum is over all unlabelled -vertex graphs .
We count in two ways the number of sequences of subgraphs of such that , . We have
where the sum extends over all unlabelled graphs on at most vertices. Since , it follows by Kelly’s Lemma that the left-hand side of this equation is reconstructible. On the other hand, the terms are also reconstructible whenever . The result follows after rearranging Equation 1. ∎
To state our results in full generality, we make the following definition.
Let be a class of graphs on vertices. We say that satisfies Kocay’s lemma if, for every graph and every sequence of graphs , where , , the sum
The following proposition gives a simple condition that is sufficient for a class of graphs to satisfy Kocay’s lemma.
Let be a class of graphs on vertices. Suppose that is reconstructible for every and for every -vertex graph . Then the class satisfies Kocay’s lemma.
The class of connected simple graphs satisfies Kocay’s lemma, since if is any connected graph and is any disconnected graph, then is reconstructible (see Bondy ). Other classes of graphs that satisfy Kocay’s lemma include planar graphs, trees and of course the class of all graphs. Our theorems apply to finite and recognisable classes of graphs satisfying Kocay’s Lemma. All the above classes of graphs are recognisable as well.
Let be a finite, recognisable class of -vertex graphs satisfying Kocay’s Lemma. In the rest of this paper, we study equations obtained by applying Kocay’s Lemma to . It is useful to view this lemma as follows. Let , be a sequence of graphs where for each . Let , i.e., is a reconstruction of , and since is recognisable, is in . Then we have
where is a constant that depends only on the sequence and the reconstruction class , i.e., it is a reconstructible parameter. In this expression, is constant (i.e., it is independent of the reconstruction class) and depends on the isomorphism class of a particular reconstruction of under consideration. Therefore, each application of Kocay’s Lemma provides a linear constraint on that all reconstructions of must satisfy.
This paper is devoted to a study of systems of such linear constraints obtained by applications of Kocay’s lemma. In particular, we study the rank of a matrix of covering numbers that we define next.
Let be a finite class of graphs on vertices. Let be a family of sequences of graphs on at most vertices. We let to be a matrix whose rows are indexed by the sequences and whose columns indexed by the distinct isomorphism classes of graphs in . The entries of are the covering numbers defined by , where and .
3 On the rank of a matrix obtained from Kocay’s Lemma
3.1 Large rank implies few non-reconstructible graphs
As observed earlier, for any finite class of graphs, , and the bigger the number of distinct reconstruction classes, the smaller is the number of non-reconstructible graphs. The main result of this section, Theorem 3.2, states that for any finite, recognisable class of graphs satisfying Kocay’s lemma, the number of distinct reconstruction classes is bounded from below by the rank of the matrix of covering numbers, for any family of sequences of graphs.
Let be a finite, recognisable class of -vertex graphs satisfying Kocay’s Lemma. Let be a finite family of sequences of graphs on at most vertices. Let be the corresponding matrix of covering numbers , where and (see Definition 2.14). Let be a subspace of the vector space over . We associate with the constant .
If , the result is trivial. Otherwise, let be the non-reconstructible reconstruction classes in , i.e., for all . Let be the isomorphism classes in . Let be representative graphs from .
For each , we define a vector , with its entries, which are indexed by unlabelled graphs , defined as follows:
Observe that to prove the lemma it is enough to show that the vectors satisfy the following properties:
for all , for all , ; and
the vectors in the set are non-zero and linearly independent, where .
Proof of (i): Graphs and are reconstructions of each other, and satisfies Kocay’s Lemma. Therefore, for every row of , we have,
Proof of (ii): Let the vectors in be ordered so that the corresponding graphs are ordered by non-decreasing numbers of small edges. We prove that is non-zero, and for each , the vector is non-zero and is linearly independent of , which would imply that the vectors in are linearly independent.
Let for some and . First recall that is recognisable, , and ; therefore, . In addition, since and these two graphs belong to distinct isomorphism classes within the same reconstruction class . Finally, the number of small edges is reconstructible, i.e., . Therefore,
Now consider the vectors and , where . We prove that . Since , according to the ordering of , we have . Since and are reconstructions of each other, we have .
Now, if , then
On the other hand, if , then again (since and are non-isomorphic but have the same number of edges) and (because , so and are non-isomorphic but have the same number of edges).
Now the lemma follows from . ∎
Let be a finite, recognisable class of -vertex graphs satisfying Kocay’s Lemma. Let be a family of sequences of graphs on at most vertices. If is the corresponding matrix of covering numbers associated with and , then .
Applying the Rank-Nullity Theorem, we have
It follows from Lemma 3.1 that
Now recalling the definition of , we have
which implies that . ∎
Under the hypotheses of Theorem 3.2, if then every graph in is reconstructible.
Figure 1 illustrates an application of Corollary 3.3 to the class of connected graphs on four vertices. We show six sequences of graphs (indexing rows) and the corresponding covering numbers for each of the six connected graphs on four vertices (indexing the columns). A zero in -th row and the -th column (e.g., most entries in the upper triangle) indicates that there is no way to cover the corresponding graph (indexing a column) by graphs in the corresponding sequence (indexing the row). The matrix has full rank, implying that connected graphs on four vertices are reconstructible.
3.2 The existence of matrices with optimal rank
Let be a recognizable class of -vertex graphs satisfying Kocay’s lemma. Then there exists a family of sequences of graphs with corresponding matrix of covering numbers such that .
Let be the family of all inequivalent sequences of length at most of -vertex graphs. Here we consider two sequences and to be inequivalent if for each bijection from to , there is at least one graph in for which is not isomorphic to . Since the covering numbers for sequences of length 1 in are all 0, we assume that contains only sequences of length at least 2. Let be the corresponding matrix of covering numbers. We show below that this choice for the family of sequences and its corresponding matrix of covering numbers satisfy the desired property.
For a sequence and a graph , let denote the number of tuples of subgraphs of with distinct vertex sets such that , , and . We call such covers non-overlapping. Correspondingly, we have the matrix of non-overlapping covering numbers.
Now let be a sequence in . We have the following recurrence for :
where denotes the set of all onto functions from to , and is the subsequence of consisting of , and the innermost sum is over all inequivalent sequences of length of graphs on vertices. This may be explained as follows. Each cover of by naturally corresponds to a partition of in blocks for some , so that are in the same partition if and only if graphs and have the same vertex set. We denote partitions of in blocks by onto maps from to so that the inverse image denotes the -th block. For the -th block of an onto map , the union of graphs is a graph on vertices. We denote the subsequence of with indices by . Now the cover of by the sequence is non-overlapping, and each may be covered by in ways. We do not need to consider the trivial partition of into a single block, because there is no cover of by such that all have the same vertex set. In other words, the above formula computes by partitioning the coverings according to , , and , and then counting the number of coverings in each block of the partition. Since in the formula we use onto functions instead of partitions, the same block of coverings under this partition may be counted more than once, and therefore there is factor in the formula. If sequence contains copies of a graph , copies of a graph , and so on, where are mutually non-isomorphic graphs, then .
Now we rearrange the terms and write
Thus we have expressed the non-overlapping covering numbers for a sequence of length of graphs in terms of the non-overlapping covering numbers for sequences of length at most . In the above equation, are constants independent of . Also, if , we have . Therefore, by repeatedly applying the above equation to terms containing non-overlapping covering numbers, we eventually obtain
We have written the coefficients as to emphasize that they arise from factors and that do not depend on . That is, the linear dependence of the non-overlapping covering numbers on the covering numbers is the same for all graphs (but of course depends on ). Therefore, we can write
In this manner we have shown that the rows of are in the span of the rows of . Therefore, we have
To show that the rank of is , we construct a square submatrix of as follows. Let . First, for each reconstruction class , we choose one reconstruction arbitrarily from . For each , we keep the row indexed by the sequence (say ) that is equivalent to the sequence , where the vertices of may be ordered arbitrarily, and we keep the column indexed by . We delete all other rows and columns of . We show that has full rank, which will imply that .
We define a partial order on so that if there exists a bijection from to such that for each in , the graph is isomorphic to a subgraph of .
First we verify that the above relation is a partial order on . The reflexivity and the transitivity are straightforward to verify. We now verify antisymmetry. Let be a bijection as in the above paragraph. Therefore, for each , we have . Let be a similar bijection from to . Therefore, the bijective composition from to is such that for all in , we have is isomorphic to a subgraph of , implying that . Now observe that , since is a bijection from onto itself. Therefore, we must have for all , implying that and are isomorphic for all . In other words, .
We sort the rows and the columns of so that if , then is to the right of , and the row corresponding to the sequence is above the row corresponding to the family .
Now if then , therefore, the matrix is upper-triangular. Also, for all . Therefore, has full rank; in fact is equal . Since the class is recognizable and satisfies Kocay’s lemma, Theorem 3.2 is applicable. Therefore,
which implies the claim for our choice of , and the corresponding matrix . ∎
We provide another simple but non-trivial example in directed graphs, which are in general not reconstructible. Figure 2 illustrates a matrix of covering numbers for directed graphs on 3 vertices, with no multi-arcs or loops. Observe that there are 7 distinct graphs in 4 reconstruction classes: and are reconstructible; belong to the same reconstruction class; belong to the same reconstruction class. The figure shows 4 rows of the matrix corresponding to 4 graph sequences. The rank of the matrix is 4, which is also the number of reconstruction classes. It is possible to verify that the rank cannot be improved by adding more sequences of graphs.
We would like to thank Hiệp Hàn for useful discussions, and the anonymous referees for bringing our attention to the works of Kocay  and Mnukhin . The first author is grateful to Yoshiharu Kohayakawa for hosting him at Universidade de São Paulo, and would like to thank Orlando Lee for helpful discussions at an early stage of this work.
- Supported in part by NSF grants CCF-0915929 and CCF-1115703.
- Supported by CNPq grant 151782/2010-5 and by MaCLinC Project at Universidade de São Paulo.
- Observe that we are defining graphs using triples because multiple edges are allowed.
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