An algebraic formulation of the graph reconstruction conjecture

0.39 MB·English

by  Igor C. Oliveira

An algebraic formulation of the graph reconstruction conjecture
Igor C. Oliveira ∗ Bhalchandra D. Thatte †
Columbia University Universidade de S˜ao Paulo
[email protected] [email protected]
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2 January 18, 2013

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Abstract
1

The graph reconstruction conjecture asserts that a finite simple graph on at least 3 vertices
]
O can be reconstructed up to isomorphism from its deck - the collection of its vertex-deleted
C subgraphs. Kocay’s Lemma is an important tool in graph reconstruction. Roughly speaking,
given the deck of a graph G and any finite sequence of graphs, it gives a linear constraint that
.
h every reconstruction of G must satisfy.
t
a Let ψ(n) be the number of distinct (mutually non-isomorphic) graphs on n vertices, and
m let d(n) be the number of distinct decks that can be constructed from these graphs. Then the
[ difference ψ(n) d(n) is a measure of how many graphs cannot be reconstructed from their
−
decks. In particular, the graph reconstruction conjecture holds for n-vertex graphs if and only
1
if ψ(n) = d(n). We give a framework based on Kocay’s lemma to study this discrepancy.
v
1 We prove that if M is a matrix of covering numbers of graphs by sequences of graphs then
2 d(n) rankR(M). In particular, all n-vertex graphs are reconstructible if one such matrix has
1 rank≥ψ(n). Tocomplementthisresult,weprovethatitispossibletochooseafamilyofsequences
4
ofgraphssuchthatthecorrespondingmatrixM ofcoveringnumberssatisfiesd(n)=rankR(M).
.
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0
3 1 Introduction
1
:
v The graph reconstruction conjecture was proposed by Ulam [11] and Kelly [5]. Informally, it
i
X states that if two finite, undirected, simple graphs on at least 3 vertices have the same collection
r (multi-set or deck) of unlabelled vertex-deleted subgraphs, then the graphs are isomorphic; in other
a
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 [8] for all undirected, finite, simple graphs on eleven
or fewer vertices. In addition, it has been proven for many particular classes of undirected, finite,
simple graphs, such as regular graphs, disconnected graphs and trees (Kelly [6]). In fact, Bollob´as
[2] showed that it holds for almost all finite, simple, undirected graphs. On the other hand, a
similar conjecture does not hold for directed graphs: Stockmeyer [9, 10] constructed a number of
∗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 Sa˜o Paulo.
1
infinite families of non-reconstructible directed graphs. For a more comprehensive introduction to
the problem, we refer to a survey by Bondy [3]. For the standard graph theoretic terminology not
defined here, we refer to West [12].
Kelly’s Lemma [6] is one of the most useful results in graph reconstruction. Let s(F,G) denote
the number of subgraphs of G isomorphic to F. Kelly’s lemma states that for v(F) < v(G),
the parameter s(F,G) is reconstructible, in the sense that if G and G(cid:48) have the same deck then
s(F,G(cid:48)) = s(F,G). Several propositions in graph reconstruction rely on this useful lemma.
Kocay’sLemma[7]allowsus,tosomeextent,toovercometherestrictionv(F) < v(G)inKelly’s
lemma. It provides a linear constraint on s(F,G) that must be satisfied by every reconstruction of
G. Informally, it says that, if = (F ,...,F ) is a sequence of graphs, each of which has at most
1 m
F (cid:80)
v(G) 1 vertices, then there are constants c( ,H) such that the value of the sum c( ,H)
− F H F ·
s(H,G) is reconstructible, where the sum is taken over all unlabelled n-vertex graphs H. Roughly
speaking, the constant c( ,H) counts the number of ways to cover the graph H by graphs in the
F
sequence .
F
Kocay’s Lemma has been used to show several interesting results in graph reconstruction. For
instance,bycarefullyselectingthesequence ,itispossibletogiveasimpleproofthatdisconnected
F
graphsarereconstructible. Inaddition,itcanbeusedtoshowthatthenumberofperfectmatchings,
the number of spanning trees, the characteristic polynomial, the chromatic polynomial, and many
other parameters of interest are reconstructible; see Bondy [3].
It is natural to wonder whether even more restrictions may be imposed on the reconstructions
of G by applications of Kocay’s Lemma. Recall that it is possible to use different sequences of
graphs in each such invocation of Kocay’s lemma, and as explained before, for each such sequence
we get a linear constraint that the reconstructions of G must satisfy. By analysing such equations
one would expect to obtain a wealth of information about the structure of any reconstruction of G
(perhaps enough equations may even allow us to conclude that G is reconstructible). In this paper
we investigate how much information one can obtain by setting up such equations.
WeprovethattheequationsobtainedbyapplyingKocay’sLemmatothedeckofagraphGusing
distinctsequencesofgraphsprovideimportantinformationnotonlyaboutthereconstructionsofG,
butalsoaboutthetotalnumberofnon-reconstructiblegraphsonnvertices. Moreformally,letd(n)
be the number of distinct decks obtained from n-vertex graphs. We show that if M is the matrix
of coefficients corresponding to these equations, then d(n) rankR(M), 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 n vertices are reconstructible. In addition, we
give a proof that there exist d(n) sequences of graphs , ,..., , with corresponding matrix
1 2 d(n)
F F F
M of covering numbers, such that rankR(M) = d(n). In other words, if the graph reconstruction
conjectureholdsforgraphswithnvertices, thenthereisacorrespondingfull-rankmatrixcertifying
this fact.
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.
2 Preliminaries
In this paper, we consider general finite graphs - undirected graphs, directed graphs, hyper-
graphs, graphs with or without multiple edges, and with or without loops. We take the vertex set
2
of a graph to be a finite subset of N. We write V(k) for the family of k-element subsets of a set V.
Definition 2.1 (Graphs). A hypergraph G is a triple (V,E,φ), where V is its vertex set (also
calledground set, andwritten asV(G))andE isits setofhyperedges (writtenasE(G)), anda map
φ : E 2V 1. Anundirectedgraph Gisahypergraphwiththerestrictionthatφ : E V(1) V(2);
→ \∅ → ∪
in this case we call a hyperedge e an edge (if φ(e) = 2) or a loop (if φ(e) = 1). A directed graph G
| | | |
is a triple (V,E,ψ), where V is its vertex set and E is the set of its arcs, and a map ψ : E V V.
→ ×
The first element of ψ(e) is called the tail of the arc e, and the second element of ψ(e) is called the
head of e. We denote the set of all finite graphs (including hypergraphs, undirected graphs and
directed graphs) by ∗.
G
Remark 2.2. 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 G and H be two graphs. We say that G and H are
isomorphic (written as G = H) if there are one-one maps f : V(G) V(H) and g : E(G) E(H)
∼
→ →
such that an edge e and a vertex v are incident in G if and only the edge g(e) and the vertex f(v)
are incident in H. Additionally, in the case of directed graphs, a vertex v is the head (or the tail)
of an arc e if and only if f(v) is the head (or, respectively, the tail) of g(e). The isomorphism class
of a graph G, denoted by G/=, is the set of graphs isomorphic to G.
∼
Definition 2.4. 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). LetGbegraphandletv beavertexofG. Theinducedsubgraph
of G obtained by deleting v and all edges incident with v is called a vertex-deleted subgraph of G,
and is written as G v. We say that H is a reconstruction of G (written as H G) if there is a
− ∼
one-one map f : V(G) V(H) such that for all v V(G), the graphs G v and H f(v) are
→ ∈ − −
isomorphic. The relation is an equivalence relation. We say that a graph G is reconstructible if
∼
every reconstruction of G is isomorphic to G (i.e., if H G implies H = G). A parameter t(G) is
∼
∼
said to be reconstructible if t(H) = t(G) for all reconstructions H of G. Let be a class of graphs.
C
We say that is recognisable if, for any G , every reconstruction of G is in . Furthermore, we
C ∈ C C
say that is reconstructible if every graph G is reconstructible.
C ∈ C
Example 2.6. LetG(V,E,φ)beahypergraph. Thenumberofedgesincidentwithallvertices(i.e.,
edges e E such that φ(e) = V, which we call big edges), is not a reconstructible parameter. For
∈
example, if Gk is a graph obtained from G by adding k new edges e ,e ,...,e and making them
1 2 k
incident with all vertices in V, then Gk is a reconstruction of G. In this sense, no hypergraphs are
reconstructible, and each hypergraph has infinitely many mutually non-isomorphic reconstructions.
If G is a graph in class , then is not recognisable if for some k, the graph Gk is not in ; and
C C C
is not finite if graphs Gk are all in . On the other hand, the number of small edges, i.e., edges
C C
e E such that φ(e) = V, is a reconstructible parameter.
∈ (cid:54)
In view of the above example, we will always use ∗ for the set of all graphs, for the set of all
G G
graphs without big edges, and for the set of n-vertex graphs without big edges. A class will
n n
G C
always be a subset of . We will use the following slightly restrictive definitions for some other
n
G
reconstruction terms.
1Observe that we are defining graphs using triples because multiple edges are allowed.
3
Definition 2.7. A graph G in is reconstructible if it is reconstructible modulo big edges, i.e., if G(cid:48)
G
is a reconstruction of G and G(cid:48) , then G(cid:48) is isomorphic to G. A subclass of is recognisable if
∈ G C G
foreachgraphGin , eachreconstructionofGin isalsoin . Asubclass of isreconstructible
C G C C G
if each graph in is reconstructible (modulo big edges).
C
Example 2.8. Disconnected undirected graphs on 3 or more vertices are recognisable and re-
constructible. 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. [1]).
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
∼ n ∼ ∼ n ∼
G G C C
an isomorphism class of graphs. But sometimes we abuse the notation slightly, e.g., if a quantity
is invariant over an isomorphism class H, then in the same context we may also use H to mean
a representative graph in the class. Similarly, we denote various reconstruction classes by / ,
G ∼
/ , / , / , and so on. Note that equivalence classes of any class of graphs under are
n n
G ∼ C ∼ C ∼ ∼
refined by =; in particular, / /= , and equality holds if and only if the class is
∼ n n ∼ n
|C ∼| ≤ |C | C
reconstructible. We will refer to reconstruction classes of (i.e., members of / ) by R ,R ,...,
n n 1 2
C C ∼
and isomorphism classes of R (i.e., members of R /=) by R ,R ,....
i i ∼ i,1 i,2
Given graphs G and H, the number of subgraphs of G isomorphic to H is denoted by s(H,G).
The following two subgraph counting lemmas are important results about the reconstructibility of
the parameter s(H,G).
Lemma 2.9 (Kelly’s lemma, [6]). Let H be a reconstruction of G. If F is any graph such that
v(F) < v(G), then s(F,G) = s(F,H).
Definition 2.10. Let G be a graph and let := (F ,F ,...,F ) be a sequence of graphs. A cover
1 2 m
F
of G by is a sequence (G ,G ,...,G ) of subgraphs of G such that G = F , 1 i m, and
1 2 m i ∼ i
(cid:83) F ≤ ≤
G = G. The number of covers of G by is denoted by c( ,G).
i
F F
Lemma 2.11 (Kocay’s lemma, [7]). Let G be a graph on n vertices. For any sequence of graphs
:= (F ,F ,...,F ), where v(F ) < n, 1 i m, the parameter
1 2 m i
F ≤ ≤
(cid:88)
c( ,H)s(H,G)
F
H
is reconstructible, where the sum is over all unlabelled n-vertex graphs H.
Proof. We count in two ways the number of sequences (G ,...,G ) of subgraphs of G such that
1 m
G = F , 1 i m. We have
i ∼ i
≤ ≤
m
(cid:89) (cid:88)
s(F ,G) = c( ,X)s(X,G), (1)
i
F
i=1 X
where the sum extends over all unlabelled graphs X on at most n vertices. Since v(F ) < n, it
i
follows by Kelly’s Lemma that the left-hand side of this equation is reconstructible. On the other
hand, the terms c( ,X)s(X,G) are also reconstructible whenever v(X) < n. The result follows
F
after rearranging Equation 1.
4
To state our results in full generality, we make the following definition.
Definition 2.12. Let beaclassofgraphsonnvertices. Wesaythat satisfies Kocay’s lemma
n n
C C
if, for every graph G and every sequence of graphs = (F ,F ,...,F ), where v(F ) < n,
n 1 2 m i
∈ C F
1 i m, the sum
≤ ≤ (cid:88)
c( ,H)s(H,G)
F
H∈Cn/∼=
is reconstructible.
The following proposition gives a simple condition that is sufficient for a class of graphs to
n
C
satisfy Kocay’s lemma.
Proposition 2.13. Let beaclassofgraphsonnvertices. Supposethats(H,G)isreconstructible
n
C
for every G and for every n-vertex graph H / . Then the class satisfies Kocay’s lemma.
n n n
∈ C ∈ C C
Proof. Let G . Let := (F ,F ,...,F ) be any sequence of graphs such that v(F ) < n,
n 1 2 m i
∈ C F
1 i m. We write the R.H.S. of Equation 1 as
≤ ≤
(cid:88) (cid:88)
c( ,H)s(H,G)+ c( ,H)s(H,G),
F F
H∈Cn/∼= H∈/Cn/∼=
where the second summation is reconstructible. Now we rearrange the terms in Equation 1 to
(cid:80)
obtain c( ,H)s(H,G).
H∈Cn/∼= F
The class of connected simple graphs satisfies Kocay’s lemma since if G is any connected graph
and H is any disconnected graph, then s(H,G) is reconstructible (see Bondy [3]). 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 n-vertex graphs satisfying Kocay’s Lemma. In
n n
C ⊆ G
the rest of this paper, we study equations obtained by applying Kocay’s Lemma to . It is useful
n
C
to view this lemma as follows. Let := (F ,...,F ), be a sequence of graphs where v(F ) < n
1 m i
F
for each 1 i m. Let G,G(cid:48) R / , i.e., G(cid:48) is a reconstruction of G, and since is
n n
≤ ≤ ∈ ∈ C ∼ C
recognisable, G(cid:48) is in . Then we have
n
C
(cid:88)
c( ,H)s(H,G(cid:48)) = k ,
F,R
F
H∈Cn/∼=
wherek isaconstantthatdependsonlyonthesequence andthereconstructionclassR,i.e.,it
F,R
F
is a reconstructible parameter. In this expression, c( ,H) is constant (i.e., it is independent of the
F
reconstruction class) and s(H,G(cid:48)) depends on the isomorphism class of a particular reconstruction
G(cid:48) of G under consideration. Therefore, each application of Kocay’s Lemma provides a linear
constraint on s(H,G(cid:48)) that all reconstructions G(cid:48) of G 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.
5
Definition 2.14. Let be a finite class of graphs on n vertices. Let F = ( , ,..., ) be a
n 1 2 l
family of sequences of Cgraphs on at most n − 1 vertices. We let MF,Cn/∼= ∈FR|FF|×|Cn/∼=|Fto be a
matrix whose rows are indexed by the sequences ,i = 1,2,...,l and whose columns indexed by
i
F
the distinct isomorphism classes of graphs in Cn. The entries of MF,Cn/∼= are the covering numbers
defined by c( ,H), where F and H /=.
n ∼
F F ∈ ∈ C
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
n n n ∼
C |C ∼| ≤ |C |
number of distinct reconstruction classes, the smaller is the number of non-reconstructible graphs.
The main result Theorem 3.2 of this section states that for any finite, recognisable class of graphs
satisfying Kocay’s lemma, the number of distinct reconstruction classes is bounded below by the
rank of the matrix of covering numbers for any system of sequences of graphs.
Let be a finite, recognisable class of n-vertex graphs satisfying Kocay’s Lemma. Let F be a
n
C
finite family of sequences of graphs on at most n−1 vertices. Let MF,Cn/∼= be the corresponding
matrix of covering numbers c( ,H), where F and H /= (see Definition 2.14). Let
n ∼
W = {x ∈ R|Cn/∼=| | MF,Cn/∼=F· x ≡ 0} be aFsu∈bspace of th∈e vCector space R|Cn/∼=| over R. We
associate with the constant α( ) := /= / .
n n n ∼ n
C C |C |−|C ∼|
Lemma 3.1. dim(W) α( ).
n
≥ C
Proof. Ifα( ) = 0,theresultistrivial. Otherwise,letR ,...,R / bethenon-reconstructible
n 1 s n
C ∈ C ∼
reconstruction classes in , i.e., r := R /= > 1 for all i 1,2,...,s . Let R ,j 1,2,...,r
n i i ∼ i,j i
C | | ∈ { } ∈ { }
be the isomorphism classes in R ,i 1,2,...,s . Let G be representative graphs from R .
i i,j i,j
ForeachGi,j, wedefineavector∈w{i,j R|Cn/∼=}|, withitsentries, whichareindexedbyunlabelled
∈
graphs H /=, defined as follows:
n ∼
∈ C
wi,j(H) := s(H,G ) s(H,G ), whereH /= .
i,j i,1 n ∼
− ∈ C
Observe that to prove the lemma it is enough to show that the vectors wi,j satisfy the following
properties:
(i) for all i 1,2,...,s , for all j 1,2,...,r , wi,j W; and
i
∈ { } ∈ { } ∈
(ii) the vectors in the set U := wi,j 1 i s, 2 j r are non-zero and linearly
i
{ | ≤ ≤ ≤ ≤ }
independent, where U = α( ).
n
| | C
Proof of (i): Graphs G and G are reconstructions of each other, and satisfies Kocay’s
i,j i,1 n
C
Lemma. Therefore, for every row MF of MF,Cn/∼=, we have,
(cid:88) (cid:88)
c( ,H)s(H,G ) = c( ,H)s(H,G )
i,j i,1
F F
H∈Cn/∼= H∈Cn/∼=
(cid:88)
∴ M wi,j = c( ,H)(s(H,G ) s(H,G )) = 0.
F i,j i,1
· F −
H∈Cn/∼=
Therefore, MF,Cn/∼=·wi,j = 0.
6
Proof of (ii): Let the vectors in U be ordered u1,u2,...,uα(Cn) so that the corresponding graphs
are ordered by non-decreasing numbers of small edges. We prove that u1 is non-zero, and for each
k 2,...,α( ) , the vector uk is non-zero and is linearly independent of u1,u2,...,uk−1, which
n
∈ { C }
would imply that the vectors in U are linearly independent.
Let u(cid:96) = wi,j for some i 1,2,...,s and j 2,...,r . First recall that is recognisable,
i n
∈ { } ∈ { } C
R / , and G R /=; therefore, G /=. In addition, G (cid:29) G since j 2 and these
i n i,j i ∼ i,j n ∼ i,j i,1
∈ C ∼ ∈ ∈ C ≥
two graphs belong to distinct isomorphism classes within the same reconstruction class R . Finally,
i
the number of small edges is reconstructible, i.e., e(G ) = e(G ). Therefore,
i,j i,1
u(cid:96)(G ) = wi,j(G ) = s(G ,G ) s(G ,G ) = 1 0 = 1.
i,j i,j i,j i,j i,j i,1
− −
Nowconsiderthevectorsuk = wi(cid:48),j(cid:48) andu(cid:96) = wi,j,where1 k < (cid:96). Weprovethatuk(G ) = 0.
i,j
≤
Since k < (cid:96), according to the ordering of U, we have e(G ) e(G ). Since G and G are
i(cid:48),j(cid:48) i,j i(cid:48),j(cid:48) i(cid:48),1
≤
reconstructions of each other, we have e(G ) = e(G ).
i(cid:48),j(cid:48) i(cid:48),1
Now, if e(G ) < e(G ), then
i(cid:48),j(cid:48) i,j
uk(G ) = wi(cid:48),j(cid:48)(G ) = s(G ,G ) s(G ,G ) = 0 0 = 0.
i,j i,j i,j i(cid:48),j(cid:48) i,j i(cid:48),1
− −
On the other hand, if e(G ) = e(G ), then again s(G ,G ) = 0 (since G and G are
i(cid:48),j(cid:48) i,j i,j i(cid:48),j(cid:48) i,j i(cid:48),j(cid:48)
non-isomorphic but have the same number of edges) and s(G ,G ) = 0 (because j > 1, so G
i,j i(cid:48),1 i,j
and G are non-isomorphic but have the same number of edges).
i(cid:48),1
Now the lemma follows from α( ) := /= / = (cid:80)s (r 1) = U .
Cn |Cn ∼|−|Cn ∼| i=1 i− | |
Theorem 3.2. Let be a finite, recognisable class of n-vertex graphs satisfying Kocay’s Lemma.
n
C
Let F be a family of sequences of graphs on at most n−1 vertices. If MF,Cn/∼= is the corresponding
matrix of covering numbers associated with F and Cn, then |Cn/∼| ≥ rankR(MF,Cn/∼=).
Proof. Applying the Rank-Nullity Theorem, we have
dim(W)+rankR(MF,Cn/∼=) = |Cn/∼=|.
It follows from Lemma 3.1 that
α(Cn)+rankR(MF,Cn/∼=) ≤ dim(W)+rankR(MF,Cn/∼=) = |Cn/∼=|.
Now recalling the definition of α( ), we have
n
C
|Cn/∼=|−|Cn/∼|+rankR(MF,Cn/∼=) ≤ |Cn/∼=|,
which implies that |Cn/∼| ≥ rankR(M)F,Cn/∼=.
Corollary 3.3. Under the hypotheses of Theorem 3.2, if rankR(MF,Cn) = |Cn/∼=| then every graph
in is reconstructible.
n
C
Figure 1 illustrates an application of Corollary 3.3 to the class of connected graphs on four
vertices.
7
G G G G G G
1 2 3 4 5 6
=
F1 2 3 0 0 0 0
,
=
F2 6 6 0 0 0 0
, ,
=
F3 0 0 1 0 0 0
,
=
F4 36 36 24 24 0 0
, , ,
=
F5 150 150 240 240 120 0
, , , ,
=
F6 540 540 1536 1536 1800 720
, , , , ,
Figure 1: A full-rank matrix M of covering numbers c( ,G ) providing a proof through Corollary
i j
F
3.3 that all connected graphs on four vertices are reconstructible.
3.2 The existence of matrices with optimal rank
Theorem 3.4. Let be a recognizable class of n-vertex graphs satisfying Kocay’s lemma. Then
n
C
there exists a family of sequences F graphs with the corresponding matrix MF,Cn/∼= of covering
numbers such that rankR(MF,Cn/∼=) = |Cn/∼|.
Proof. LetFbethefamilyofallinequivalentsequencesoflengthatmostnof(n 1)-vertexgraphs.
−
Here we consider two sequences and to be inequivalent if for each bijection f from to
i j i
F F F
, there is at least one graph F in for which f(F) is not isomorphic to F. Since the covering
j i
F F
numbers for sequences of length 1 in F are all 0, we assume that F contains only sequences of length
at least 2. Let MF,Cn/∼= 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 G, let c∗( ,G) denote the number of tuples (G ,G ,...,G ) of
1 2 m
F F (cid:83)
subgraphs of G with distinct vertex sets such that G = F , 1 i m, and G = G. We call such
i ∼ i i
≤ ≤
covers non-overlapping. Correspondingly, we have the matrix M∗ of non-overlapping covering
F,Cn/∼=
numbers.
Now let := (F ,F ,...,F ) be a sequence in F. We have the following recurrence for c( ,G):
1 2 (cid:96)
F F
8
(cid:96) k
(cid:88) (cid:88) (cid:88) (cid:89)
c( ,G) = γ( )c∗( ,G) c( ,H ),
P−1(i) i
F H H F|
k=2P∈P(cid:96)kH:=(H1,H2,...,Hk) i=1
where k denotes the set of all onto functions from 1,2,...,(cid:96) to 1,2,...,k , and is
P(cid:96) { } { } F|P−1(i)
the subsequence of consisting of F ;j P−1(i), and the innermost sum is over all inequivalent
j
F ∈
sequences of length k of graphs on (n 1) vertices. This may be explained as follows. Each
H −
cover (G ,G ,...,G ) of G by naturally corresponds to a partition of 1,2,...,(cid:96) in k blocks
1 2 (cid:96)
F { }
for some k [2..(cid:96)], so that i,j are in the same partition if and only if graphs G and G have the
i j
∈
same vertex set. We denote partitions of 1,2,...,(cid:96) in k blocks by onto maps P from 1,2,...,(cid:96)
{ } { }
to 1,2,...,k so that the inverse image P−1(i) denotes the i-th block. For the i-th block P−1(i)
{ }
of an onto map P, the union of graphs G ;j P−1(i) is a graph H on n 1 vertices. We
j i
∈ −
denote the subsequence of with indices j P−1(i) by . Now the cover of G by the
P−1(i)
F ∈ F|
sequence := (H ,H ,...,H ) is non-overlapping, and each H may be covered by F ;j P−1(i)
1 2 k i j
H ∈
in c( ,H ) ways. We do not need to consider the trivial partition of 1,2,...,(cid:96) into a single
P−1(i) i
F| { }
block, because there is no cover (G ,G ,...,G ) of G by such that all G have the same vertex
1 2 (cid:96) i
F
set. In other words, the above formula computes c( ,G) by partitioning the coverings according
F
to k, P, and , and then counting the number of coverings in each block of the partition. Since
H
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 γ( ) in the formula. If
H
sequence contains k copies of a graph Γ , k copies of a graph Γ , and so on, where Γ are
1 1 2 2 i
mutually nHon-isomorphic graphs, then γ( ) = ((cid:81) k !)−1.
H i i
Now we rearrange the terms and write
(cid:96)−1 k
(cid:88) (cid:88) (cid:88) (cid:89)
c∗( ,G) = c( ,G) γ( )c∗( ,G) c( ,H ).
P−1(i) i
F F − H H F|
k=2P∈P(cid:96)kH:=(H1,H2,...,Hk) i=1
Thus we have expressed the non-overlapping covering numbers for a sequence of length (cid:96) of graphs
intermsofthenon-overlappingcoveringnumbersforsequencesoflengthatmost(cid:96) 1. Intheabove
−
equation,c( ,H )areconstantsindependentofG. Also,if(cid:96) = 2,wehavec∗( ,G) = c( ,G).
P−1(i) i
F| F F
Therefore, byrepeatedlyapplyingtheaboveequationtotermscontainingnon-overlappingcovering
numbers, we eventually obtain
(cid:88)
c∗( ,G) = β ( (cid:48))c( (cid:48),G).
F
F F F
F(cid:48)
We have written the coefficients as β ( (cid:48)) to emphasize that they arise from factors c( ,H )
F P−1(i) i
F F|
and γ( ) that do not depend on G. That is, the linear dependence of the non-overlapping covering
H
numbersonthecoveringnumbersisthesameforallgraphs(butofcoursedependson ). Therefore,
F
we can write
(cid:88)
c∗( , ) = β ( (cid:48))c( (cid:48), ).
F
F − F F −
F(cid:48)
In this manner we have shown that the rows of MF∗,Cn/∼= are in the span of the rows of MF,Cn/∼=.
Therefore, we have
rankR(MF∗,Cn/∼=) ≤ rankR(MF,Cn/∼=).
9
To show that the rank of M∗ is / , we construct a square submatrix K of M∗
F,Cn/∼= |Cn ∼| F,Cn/∼=
as follows. Let R ,i = 1,2,... := / . First, for each reconstruction class R ,i = 1,2,...,
i n i
{ } C ∼
we choose one reconstruction G arbitrarily from R /=. For each i = 1,2,..., we keep the row
i i ∼
indexed by the sequence (say ) that is equivalent to the sequence (G v,v V(G )), where
i i i
F − ∈
the vertices of G may be ordered arbitrarily, and we keep the column indexed by G . We delete
i i
all other rows and columns of M∗ . We show that K has full rank, which will imply that
F,Cn/∼=
rankR(MF∗,Cn/∼=) ≥ rankR(K) = |Cn/∼|.
We define a partial order on / so that R R if there exists a bijection f from V(G )
n i j i
≤ C ∼ ≤
to V(G ) such that for each v in V(G ), the graph G v is isomorphic to a subgraph of G f(v).
j i i j
− −
First we verify that the above relation is a partial order on / . The reflexivity and the
n
≤ C ∼
transitivity are straightforward to verify. We now verify antisymmetry. Let f be a bijection as
in the above paragraph. Therefore, for each v V(G ), we have e(G v) e(G f(v)). Let
i i j
∈ − ≤ −
g be a similar bijection from V(G ) to V(G ). Therefore, the bijective composition g f from
j i
◦
V(G ) to V(G ) is such that for all v in V(G ), we have G v is isomorphic to a subgraph of
i i i i
−
G (g f)(v), implying that e(G v) e(G f(v)) e(G (g f)(v)). Now observe that
i i j i
(cid:80) − ◦ (cid:80) − ≤ − ≤ − ◦
e(G v) = e(G (g f)(v)), since g f is a bijection from V(G ) onto itself. Therefore,
v i− v i− ◦ ◦ i
we must have e(G v) = e(G f(v)) for all v V(G ), implying that G v and G f(v) are
i j i i j
− − ∈ − −
isomorphic for all v V(G ). In other words, R = R .
i i j
∈
We sort the rows and the columns of K so that if R < R , then G is to the right of G , and
i j j i
the row corresponding to the sequence is above the row corresponding to the family .
i j
F F
Now if c∗( ,G ) > 0 then R < R , therefore, the matrix K is upper-triangular. Also,
i j i j
F
c∗( ,G ) > 0 for all G . Therefore, K has full rank; in fact rank(K) is equal / . Since
i i i n
F |C ∼|
the class is recognizable and satisfies Kocay’s lemma, Theorem 3.2 is applicable. Therefore,
n
C
|Cn/∼| = rankR(K) ≤ rankR(MF∗,Cn/∼=) ≤ rankR(MF,Cn/∼=) ≤ |Cn/∼|,
which implies the claim for our choice of F, and the corresponding matrix MF,Cn/∼=.
Example 3.5. We show another small but non-trivial example in directed graphs, which are in
generalnotreconstructible. Figure2illustratesamatrixofcoveringnumbersfordirectedgraphson
3 vertices, with no multi-arcs or loops. Observe that there are 7 distinct graphs in 4 reconstruction
classes: G and G are reconstructible; G ,G ,G belong to the same reconstruction class; G ,G
1 2 3 4 5 6 7
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.
4 Discussion
Inthispaperwehavedescribedanalgebraicformulationofthegraphreconstructionconjecture.
Our results show that if this conjecture is true then, at least in principle, it may be proven using
equations obtained from Kocay’s lemma, and we believe that further investigation of this approach
may be fruitful. For example, it will be interesting to prove that trees are reconstructible using
the approach of this paper. On the other hand, existence of non-reconstructible directed graphs
(particularly tournaments) and hypergraphs may also be proved by an algebraic approach based
on Kocay’s lemma.
10