155
Internat. J. Math. & Math. Sci.
(1987) 155-162
Vol. i0 No.
ON THE RECONSTRUCTION OF THE MATCHING POLYNOMIAL
AND THE RECONSTRUCTION CONJECTURE
E.J. FARRELL and S.A. WAHID
Department of Mathematics
The University of the West Indies
St. Augustine, Trinidad
(Received May 30, 1984 and in revised form May 31, 1985)
ABSTRACT.
both node and edge reconstructable.
(ii)
given.
It is shown that the matching polynomial is
(i)
Two results are proved.
Moreover a practical method of reconstruction is
A technique is given for reconstructing a graph from its node-deleted and
This settles one part of the Reconstruction Conjecture.
edge-deleted subgraphs.
KEY WORDS AND PHRASES. Matching, perfect matching, matching polynomial, matching matrix, Reconstruction Conjecture, edge reconstruction, node reconstruction.
1980 AMS SUBJECT CLASSIFICATION CODE. 05A99, 05C99.
I.
INTRODUCTION.
The graphs considered here will be finite and will have no loops.
a graph.
The
matching polynomial
of
G
w
and
w
respectively in
a
k
wlP-2kw2 k,
(l.l)
are indeterminates (or weights) associated with each node and edge
2
G, a
G
is the number of matchings in
k
tion is taken over all the k-matchings in
with
k edges and the summa-
G.
The famous Reconstruction Conjecture (see Harary [2]) is the following.
a graph with
be such
G
has been defined (see Farrell [i]) as
m(G)
where
Let
p > 2 nodes and if the
G
then the entire graph
node-deleted subgraphs.
refer the reader to [2].
deck (i.e.
the
p
subgraphs
G
v
i)
G
If
is
is given,
can be reconstructed, uniquely up to isomorphism, from these
For an interesting historical account of this conjecture,
we
An analogous form of this conjuecture, with "node" replaced
by "edge" is called the Edge-Reconstruction Conjecture.
m(G).
In this paper we will first concern ourselves with the reconstruction of
We will therefore answer the following question.
can
m(G)
question.
be found?
G is given,
Dually, we will also give an answer to the edge version of this
If the answer is yes, we will say that
reconstructible).
Suppose that the deck of
m(G)
is reconstructible (node-
In the case of the edge version of the question, we say that
m(G)
is edge-reconstructible.
It is important to know whether or not
this.
If
m(G)
m(G)
is reconstructible.
One reason is
is reconstructible, then any graph that is characterized by
m(G) will
E.J. FARRELL and S. A. WAHID
156
be reconstructible.
In general, the reconstruction of graph polynomials can shed some
For example, the reconstruction of the
light on the Reconstruction Conjecture itself.
In Particular,
characteristic polynomial was investigated by several authors.
Gutman
and Cvetkovic [3] investigated the reconstruction of the characteristic polynomial and
its implications to
Ulam’s Conjecture.
The existence of a reconstruction for the char-
[4].
acteristic polynomial was eventually established by Tutte
Tutte (in [4]) also es-
tablished the reconstructibility of the rank polynomial and the chromatic polynomial.
Although the reconstruction of
several graph polynomials has been established, no
For example, given the deck
practical means of reconstruction exists for any of them.
G, there is no method available for finding either the characteristic
of the graph
G.
polynomial or the chromatic polynomial of
existence results.
The reconstruction results are merely
pc
It is of interest therefore, to find
methods of recon-
structing the polynomials.
m(G)
In this article, we will not only show that
is node and edge-reconstruct-
We note that
ible, but in doing so, we will give practical methods for reconstruction.
Godsil [5] has given a result (Theorem 4.1) which essentially establishes the node-
m(G).
reconstruction of a special form of
However his result does not provide a prac-
Tutte’s
tical reconstruction technique, since he used
existence result to establish the
reconstruction of the number of perfect matchings in
G.
The problem of reconstructing a graph from a given deck has always been an inter-
esting one, because of its connection with Ulam’s Conjecture (see [3]).
We will give
a solution to this problem, using the matching polynomial of the graph.
In fact, our
technique gives all the graphs with the given deck.
first part of the Reconstruction Conjecture i.e.
be reconstructed.
Thus we will have established the
G,
given the deck of
However we are unable to say whether or not
G
G
itself can
is unique.
This
would have settled the Reconstruction Conjecture.
For the graph
v
v
2
G, with p nodes and
the graph obtained from
tained from
G
G
{e
E(G)
and the edge set,
P
V(G)
{v I,
We will denote by
G-v i,
edges, the node set will be
q
e
by removing node
e
2
q
}.
G(-) eo
V.
will denote the graph ob-
by removing the nodes at the ends of the edge
the graph obtained from
G
assume that the general graph
G
has
p
nodes and
G
e.
will be
Throughout the paper, we will
eo.
by deleting the edge
eo.
q
edges unless otherwise speci-
fied.
2.
THE NODE-RECONSTRUCTION OF
m(G).
In order to establish our main result, we will need the following lemma.
LEMMA i.
PROOF.
B
Let
G
be a graph with
p
q
nodes and
(i)
(m(G))
(ii)
(m(G))
P
edges.
Then
m(G-v i)
i
j
m(G(-)ej)
We establish (i) by showing that the two polynomials
A
m(G)
and
P
i m(G-vi)
will be
similar.
have precisely the same terms with equal coefficients.
The proof (ii)
MATCHING POLYNOMIAL AND RECONSTRUCTION CONJECTURE
wlJw
Let
k
A.
be a term of
2
S
with (j+l)
that
G
G-v
will contain the matching
r
has a matching
B
Conversely, if
that
G-v
with
(j+l) nodes and
A
Hence
S-v
k
wlJw2 k.
k
B
Hence
r
w
lj+lw2k.
Let
edges.
v
It follows
V(G).
r
w
also contains a term in
wlJw2 k,
k
nodes and
Then
Jw2k
then there exists a node
m(G)
We conclude that
A
G
Therefore
edges.
It follows that
edges.
has a term in
has a term in
nodes and
contains a term in
has a matching with
r
m(G)
Then
157
such
r
has a matching
wlJ+lw2 k.
has a term in
have the same kinds of
B
and
v
terms.
Let
We will show that the coefficients of like terms are equal.
m(G).
a term in
wlJw2
coefficient of
k
in A
will be
G
Since
wlJ-lw2k
a
contains
with coefficient
pec
We define a
k.
JakW
wlJw2
J-lw2k
Now, for each matching in
k
be
Hence the
G
with
with j-1
G-v
r
will contain the term
B
such matchings, then
k
ja
Hence the result follows.
ja
will be
k
corresponding matchings in the graphs
nodes, there will be exactly
nodes.
A
Then the corresponding term in
a
Therefore the coefficients of like terms are equal.
k.
The proof of (ii) is similar.
mZching
to be a matching which contains edges only.
G
in
G
we will denote the number of perfect matchings in
coefficient of the term independent of
in
w
by
y(G).
Clearly
y(G)
is the
m(G).
THEOREM I.
P
m(G)
PROOF.
This is straightforward from the lemma, by integrating with respect to
1,2
G-v.’s)
G (i.e. the
Since we are given the deck of
i
+ y(G)
=i fm(G-vi)dw
1
we can find
Hence the summation on the RHS of Theorem
p.
blem which confronts us now is that of finding
y(G).
can be found.
(G),
we will use the concept of the matching matrix
function of
A(G)
(see Farrell and Wahid [6]).
G
be a graph with
(aij),
where
aij
w
The
d-uncO is
IA(G) I.
d(A(G))
2
2
if
i < j.
if
i > j,
if
i
If
A(G)
is
+ w2
Z
viv
E
E(G)
d(A(G-vi-vj) ).
The following lemma was established in [6].
Let
d(A(G))
G-v.
and the d-
A(G)
of
3 x 3 or smaller,
Otherwise,
wld(A(G-vi)
re(G).
be any element of the given deck.
for
The pro-
j.
defined recursively as follows.
d(A(G))
LEMMA 2.
A(G)
mocng x
nodes, we define the
w-’Jw
follows.
A(G)
p
I.
In order to give a technique
for finding
Let
m(G-v i)
w
Then we can write
G
as
E. J. FARRELL AND S. A. WAHID
158
iA(G_vi)
A(G)
A
where
contain
is the first
P
-A
p
P
1
(2 l)
to obtain the
Xip
s.
These equations can be solved
d(A(G))
in the expression for
y(G).
This coefficient will be
could then be found.
and the above analysis, we obtain the following result which esta-
From Theorem
blishes the practical
THEOREM 2.
w2P/2
The coefficient of
A will
By using Lemma 2, we can ob-
Xp_l, p
Xlp X2p,
[p/2]-f!ne equations involving these unknowns.
rain
A(G).
p-I elements in the (unknown) pth column of
boolean unknowns
p-i
A
w
m(G).
reconstruction of
THe matching polynomial is node-reconstructible.
By integrating with respect to
w
of Lemma I,
2, the expressions given in part (ii)
we get
q
m(G)
C(w I)
where
w P
(N B
j$1
f
w
I.
is a function of
We can also put
w
m(G(-)ej)
dw
+
2
The only term independent of
0 in Equation (2.2)).
2
(2.2)
C(Wl),
w
2
Therefore
in
C(w I)
m(G)
is
wlP
Hence we have the following result.
q
re(G)
THEOREM 3.
+
ji fm(G(-)ej)dw2
Suppose that we are given the q
G(-)e.
subgraphs
Hence
the RHS of Theorem 3 can be found.
wlP"
of
G.
m(G) could be found.
Then both terms on
We therefore have the
following theorem.
THEOREM 4.
The matching polynomial is reconstructible from the set of subgraphs ob-
tained by removing the pairs of nodes defined by the edges in the graph.
Again, we have given a practical means by which the reconstruction of
be carried out.
3.
m(G)
can
This is explicit from Theorem 3.
m(G).
THE EDGE-RECONSTRUCTION OF
The following lemma is the basic tool in the practical edge-reconstruction of
re(G).
q
LEMMA 3.
PROOF.
qm(G)
Let
eo
w
G.
be an edge in
classes (i) those in which
w2(m(G))
2
+
jl m(G-ej).
We can partition the matchings in
is used and (ii) those in which
e.
matchings in class (i) are all matchings in the graph
G-e..
(ii) are matchings in the graph
m(G)
By summing over the
q
w
edges in
q
jl
m(G)
w
2
2
G(-)e..
e.
G
into two
is not used.
The
The matchings in class
Hence we get
m(G(-)ej)
+
m(G-ej).
G, we get
q
jl m(G(-)ej)
The result follows by using (ii) of Lemma I.
q
+
jl m(G-ej).
D
MATCHING POLYNOMIAL AND RECONSTRUCTION CONJECTURE
Suppose that the
G-e.
edge-deleted subgraphs
q
G
of
159
Then the
are given.
q
jl= m(G-ej)
summation term
m(G)
eral expression for
q
By using the gen-
on the RHS of the equation can be found.
given in Equation (i.I), we get
ak Wl p-2kw2k
Ek kak Wl
w2
p-2k
w2
k-i
7. m(G-e ).
+ j=l
Hence we have’,
[p/2]
wlP-2kw2k
kE--0 ak(q-k)
By comparing coefficients,
a
I
m(G-e ).
J
could be found for all values of
k
Hence
k.
m(G)
Therefore Theorem 5 establishes the practical edge-reconstruction of
could be found.
m(G).
THEOREM 6.
4.
The matching polynomial is edge-reconstructible.
CONNECTIONS WITH THE RECONSTRUCTION CONJECTURE.
Lemma 2 provides a technique for constructing a graph with a given matching poly-
nomial.
m(G)
Suppose that
Then from Lemma 2
is known.
equations in the boolean unknowns
x..
we will obtain a system of
associated with off-diagonal elements of
A(G).
A(G)
could
These equations can then be solved to obtain values for the
be found.
A(G)
G
constructing
m(G)
from
Given the deck of
From
m(G),
G
G
Hence
G.
defines the graph
G
s.
xij
Hence
itself can be found.
This method of
is given in [6].
we can node-reconstruct
itself can be found.
Hence
G
m(G)
according to Theorem 2.
can be reconstructed from its deck.
We
state the result formally in the following theorem.
THEOREM 7.
If the deck of
G
is given, then the entire graph
G
can be reconstructed
(though perhaps not uniquely).
The edge analogue of this result follows by a similar argument using Theorem 6.
THEOREM 8.
If the edge-deleted subgraphs of
G
is given, then the entire graph
G
can be reconstructed (though perhaps not uniquely).
The reconstruction of the graph
A(G)
p,
G
is carried out from the mathcing matrix
in turn is defined by the solutions for the boolean unknowns
and
1,2,
p
(i#j)).
be different solutions for the
i
xij
A(G).
1,2
In practice, we have found that although there might
x..’s,
the graphs defined by the resulting matching ma-
trices are always isomorphic.
5.
ILLUSTRATIONS OF THE MAIN RESULTS.
In this section we will give two examples which illustrate Theorems 2, 6, 7 and 8.
Example
Let the following graphs be the node-deleted subgraphs of a graph
HI
H2
G.
E.J. FARRELL and S. A. WAHID
160
5
H
I
H5
4
H6
2
The following matching polynomials can be easily obtained.
m(Hl)
w15 + 13w2 + 9WlW
+
3w2 + 7WlW
+ 6w13w2 + 6WlW2
8w
m(H
w
m(H#
w
5
2
7w
2
8w
m(H4)
2
5
w15 + 3w2 + 10WlW
w15 + 3w2 + 9WlW22
w15 + 3w2 + 7WlW22.
m(H 2)
2
2
2
8w
m(H6)
7w
Therefore
f(6w
il fm(G-vi)dw
w
Using the labelled subgraph
6
5
+
+ 11w
44WlW 2
+
4
2w22 + (G)w23
+ 24w
H 6, the matching matrix
48WlW2)dw I,
Y
A(G)
defined in Equation (2.1)
is
A(G)
0
wI
0
w
-/w
I
-/w
2
Cw2
/w2
0
/w
/w
/w
w
2
2
2
2
x26w2
x36w2
I
x46w2
x56w2
I
/w 2
/w
-/w2
-/w2
-/w2
wI
0
0
-w2
-w2
0
w
/16w2
2
It can be confirmed that
w16 + (x16 + x26 + x36 + x46 + x56 + 7)w14w
d(A(G))
+
+
(5x16
+(2x16
+
4x26 + 3x36 + 4x46 + 5x56
x26
+
x36
+
x46
+ 7)w
+
2x56)w23.
+
x56
2
From Lemma 2, we get, by comparing coefficients
x16
+
x26
+
x36
+
x46
4.
5x16 + 4x26 + 3x36 + 4x46 + 5x56
17.
It is clear that the only solutions to these equations are
and
(x16,
x26, x36, x46, x56 )T
(I, 0, I, I, I) T
(x16,
x26, x36, x46, x56 )T
(i, I, i, 0, I)
T.
w
2
2
2
MATCHING POLYNOMIAL AND RECONSTRUCTION CONJECTURE
d(A(G))
By substituting into
m(G)
A(G)
We can now use
y(G)
we get in both cases
6
6.
161
Hence we obtain
llWl4W2 + 24w12w2 + 6w23.
Wl +
in order to reconstruct the graph
THe following la-
G.
belled graphs are constructed, corresponding to the two solutions obtained.
G
is ob-
tained by using the first solution.
s
51
6
2
GI
2
G2
Figure 2
It can be easily confirmed that the mapping
defined by
(2)
4; (3)
3; (4)
2; (5)
(6)
and
6
:G
G
2
such that
is an isomorphism.
(i)
Hence
Example 2
Let the following graphs be the edge-deleted
subgraphs of a graph
G1
6
G.
G2
5
4
2
G4
3
G6
G5
Figure 3
The following matching polynomials can be easily
obtained.
m(G l)
m(G3)
m(G5)
m(G
2
2
2
5w
w
2
m(G
2
6w
w
6
2
6w
2
2
m(GT)
2
7
i
Wl6 + 6Wl4W + 6w12w22 2) wi6 + 4w2 + 2w22 +
+ 6w14w +
w16 + 6Wl4W +
4)
2w22 +
w16 + 6w14w + 6w12w22 m(G6) w16 + 4w2 + 2w22 +
w16 + 6w14w + 8w12w +
i) 7w16 + 42w14w + 45w12w22 + 4w23 C(Wl, w2).
m(G
2
w
3
2
2
From Theorem 5, we get
3
kS0 me(Y-k)
w
6-2kw k
2
7w
6
+
42w14w + 45w12w22 + 4w23.
2
By comparing coefficients in this equation, we get
a
i, a
9 and
7, a 2
0
Hence the matching polynomial of
G
is
a
3
I.
7w
w
6w
w
7w
w
3
2
3
2
3
2
5;
GI--G2.
E.J. FARRELL and S. A. WAHID
162
m(G)
6
Wl +
7w14w2 + 9w12w22 + w23.
We can extend the reconstruction to
to form the following matching matrix of
-/w2
-Xl/3w2
w1
-’/w
2
G
itself by using the labelled subgraph
/w2
x/x24w2
w
/w
1
donsequently
G
bit tedious.
We will not reproduce them here.
G
2
A(G)
/w2
/w 2
x3/x 5w2 x3/’36w2
could be found.
The calculations involved in finding
(x13 x14 x15 x16 x24 x35 x36 x45 x46 )T
The graph
4
G.
By using Lemma 2, all the boolean unknowns in
can be found.
G
Hence
d(A(G))
A(G)
and
are a
One solution is
(0, 0, 0, 0, O, O, O, O, I, O)
T
defined by this solution is the following.
G:
Figure 4
REFERENCES
i.
2.
3.
4
5.
6.
FARRELL, E. J., Introduction to Matching Polynomials, J. Combinatorial Theory B,
27(1979), 75-86.
HARARY, F., A Survey of the Reconstruction Conjecture in "Graphs and Combinatorics’
R. Bari and F. Harary editors, Springer-Verlag, New York(1974) 18-28
GUTMAN, I. and CVETKOVIC, D.M., The Reconstruction Problem for Characteristic Polynomials of Graphs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No.
498
No. 541(1975) 45-48.
All the Kings Horses, in "Graph Theory and Related Topics" J A
TUTTE, W. T
Bondy and U. S. R. Murty editors, Academic Press, New York(1979) 15-33.
GODSIL, C. D., Matchings and Walks in Graphs, -L GraT.ery, (1981) 285-297.
FARRELL, E. J. and WAHID, So A., Matching Polynomials; A Matrix Approach and its
Applications, J. Franklin Inst, to appear