Internat. J. Math. & Math. Sci.
(1988) 87-94
VOL. ii NO.
87
ON STAR POLYNOMIALS, GRAPHICAL PARTITIONS AND
RECONSTRUCTION
E.J. FARRELL and C.M. DE MATAS
Department of Mathematics
University of West Indies
St. Augustine, Trinidad
(Received April 29, 1986)
It is shown that the partition of a graph can be determined from its star
ABSTRACT.
polynomial and an algorithm is given for doing so.
It is subsequently shown (as it is
well known) that the partition of a graph is reconstructible from the set of nodedeleted subgraphs.
KEY WORDS AND PHRASES. star, star polynomial, star cover, graphical partition, reconstruction.
kMS SUBJECT CLASSIFICATION CODES. 05A99, 05C99.
I.
INTRODUCTION.
The graphs here will be finite, undirected, and will have no loops or multiple
We define an
edges.
m-
(called the centre of
S
S
m
to be a tree consisting of a node of valency
joined to
m
m
other nodes.
0-6 is
A
m
a node and a
I-6Y is an edge.
Let
G
A
be a graph.
6oY-COVe (or simply, a COVe) of
determinate or
S
m
wig Wm+l;
C
and wth each star cover
is a spanning sub-
consisting of
G, an in-
S
S
m
m
2
the weight
k
k
w(c)
Then the
G
Let us associate with each m-star in
graph whose components are all stars.
602[
poZyom%o
of
ii
w
m.
1
G (relative to the given weight assignment) is
E(G;)
Zw(C),
where the summation is taken over all the star covers in
G
and
(Wl,
w
2
is
The basic results on star polynomials are given in the intro-
a general weight vector.
ductory paper by Farrell [I].
We will show that
HG
the partition of a graph
This will then be used to show that
R
G
G
can be obtained from
E(G;).
is node-reconstructible, a result that can be
established by more elementary means (see Tutte [2]).
E.J. FARRELL and C.M. DE MATAS
88
For brevity, we will ite E(G)
2.
r’s.
k
since the same weight vector
rk
Also, in partitions, we will use
used throughout the paper.
ence of
E(G;w),
for
()
Finally, we will assume that
will be
w
to denote the occur-
0, for all r
n.
STAR POLYNOMIALS AND GRAPHICAL PARTITIONS.
First of all, we state a lemma which can be easily proved.
LEMMA I.
Let
v
be a node of valency
G
in
G
Then
DEFINITION
Let
G
E(G)
of simple m-covers in
G
will have weight
c
w
6dp e0e%eewt,
a
m
Note that the term
G.
Let
p-l).
G
be a graph with
w
and its co-
will be the number
m
will also be a simple term.
P
Let
nodes
p
E(G).
(n
HG
n
,...,2
b2
b
,I
,0
b0
I,
m >
Then for
is a
in
Wm+
c
b
LEMMA 2
G
of
6%mp Am,
will be referred to as a
efficient, which will be denoted by
n
m-stars
p-m-1 isolated nodes.
It is clear that a simple m-cover in
mon6mial of this form in
S%mpl m-covEr
A
p nodes.
be a graph with
cover consisting of an m-star and
(0
()
contains
v
with centre
A
d
n
C
PROOF.
Z
r=m
m
()
m
b
r
This is straightforward.
The following lemma can be easily proved.
LEMMA 3.
Let
all the terms
for
be the largest valency of a node in
n
p-r-I
w
(0 _< r _< n)
Wr+
G
E(G)
Then
with non-zero coefficient,
i.e
contains
# 0
Cr
for
(0 ! r in)
Suppose that we put
in the above Lemma.
n
Then
G
will consist of a set of
G
Clearly then the partition of
disjoint edges and possibly isolated nodes.
will be
given by
p-2c
c
H
HG
Hence
If
I, then from Lemma 2,
we get
n
r
E
r=k+l
n
Ck
rk (kr)br bk +
b
c
n
k
Therefore
b
b’s
bk+ I, bk+2
i.e.
follows:
c
k
r.
k
r=k+l
c
For
k
n
G
is
2c
kZ=l
for k > I.
(2.2)
(which is defined in
k
and
O,
E(G))
we can find
b
and all the higher
and
b
0
as
Therefore the sum of the valencies of the
G.
is the number of edges in
n
N
nodes of
(k)br,
()b r.
can be obtained from
,b
(2.1)
).
0
E(G).
can be found from
n
(i
G
kb
k
b
+
kb
kZ=2
k
n
=2c
kE__2
kb
k.
(2.3)
STAR POLYNOMIALS, GRAPHICAL PARTITIONS AND RECONSTRUCTION
b
0
G
is the number of isolated nodes in
89
Therefore
n
b
p
0
b
iZ__l
(2.4)
i-
Our discussion yields the following theorem,
0
where
G
Let
THEOREM 1.
b
II
nodes and let
p
(n
G
b
n
2
2
obO)
b
,1
Then
p-l.
n
be a graph with
n
b
c
r
k’f)br’
Z
k
r-k+l
2c
rE__2
p
rZ__l br.
k
b
rb
<k <- no
for
r
and
n
b
For
I,
n
0
is given by Equation (2.1).
HG
For
G
yields an algorithm for obtaining
Theorem
0, the result is trivial.
n
from
E(G).
This algorithm is
illustrated in the following example:
EXAMPLE I.
E(G)
be a graph such that
G
Let
+ 6ww + 7w3w + w2w + 8wZw2 + 12w
w6
13
12
oserve
First of all, we
ple terms in
E(G)
w 6, that
w 7ww
w2w
for
k
Since the largest
from the term
6w
are
12
and
which
w
b
c
+
w
+ 3w2~
w w
2
2
has 6 nodes i.e.
G
Therefore
6
c
c
p
6.
=7
and
The sim-
I.
c
2
E(G) is 4, it follows that
occurs in
k
+
w w
123
3.
n
Now
3
b
I;
c
2
3
r
2
(2)b r
3
7
7-3
3b
3
2(6)
b
kZ2=
3
b0
Hence
p
r$1
kb
br
k
12 -2b
12-8-3
3b
6-1-4-I
0.
(312II)"
HG
G
itself from
only the simple terms in
E(G)
itself is not clearly defined.
are used to obtain
G
characterization of
is concerned.
it is not surprising that
G
E(G)
are useless as far as the
It would be interesting to investigate the
HG
is unigraphic (i.e. there is only one graph with partition
could be uniquely constructed from
PROOF.
HG
Since
’useless terms’.
Suppose that
THEOREM 2.
From Theorem I,
itself be clearly defined by the simple
G
Should
G’
terms, then it would mean that the remaining terms of
nature of these
E(G).
However there can be several graphs with the same partition.
can be obtained.
G
I.
2
It would be nice to be able to obtain
then
4.
Let
G
be unigraphic.
This follows from Theorem
Hence we have the following theorem.
HG"
Then
HG ),
G
can be constructed from
and the discussion above.
E(G).
E.J. FARRELL and C.M. DE MATAS
90
THEOREM 3. Let
E(G)
only if
I, G
from Theorem
H
E(H)
and
be two graphs with
H
and
Finally,
be equal.
E(G)
r
E(G)
I) in
(r
Hence
i, for all graphs.
c
c
E(H)
and
Then
are equal.
Conversely, suppose that
must have the same partition.
Then from Lemma 2, the coefficients
H
if and
have the same simple coefficients.
H
and
NG H
Then
nodes.
p
Suppose that the simple coefficients in
PROOF.
NG
G
E(G)
and
E(H)
and
E(H)
must
have the same
The result therefore follows.
simple coefficients.
STAR POLYNOMIALS AND RECONSTRUCTION.
3o
The following theorem is analogous to the result for circuit polynomials given in
Lemma 3 of Farrell and Grell [3],
F-polynomials (see Farrell [4]).
holds for all
from
G
We suspect that the general result
i.
i
with
by removing node
V(G)
x.
Here
G-x
denotes the graph obtained
G.
is the node set of
THEOREM 4.
aE(G)
E E(G-x;)
V(G)
3Wl
PROOF.
E A
E(G)
Let us write
j
nodes in
G
n ,j
w
j
w
n2, j
n
...Wp
2
P,J
n
Z n
w
j
,jAj Wl
w
n
,j---Iwn2,
...Wp
2
n
n2, j
nl, j
It is clear that the monomial
w
...w
2
P,j
p’j
G-x, for some node
of a cover in
x
G-x
Conversely, every cover of
adding an isolated node.
esponding monomial
a term with monomial
wlm
m.
G.
The derivative of
A.
sisting of
node
x
nl, j
isolated nodes
is removed from
isolated nodes.
G.
Since node
n2, j
Then
x
w
2
edges,..., n
G-x
nl,jAj.
E(G)
w
P,j
could be any of the
Hence the coefficient of
G
P
(p-l)
G
w
nl, j
A
has
j
stars.
-I n 2
w
2
’j...w
yields
covers con-
Suppose that
nl,j-I
isolated nodes in the
gives rise to
nl, j
by
have the same mono-
nl,
n
p
p’j
covers with one less
in
From Equation (3.1) it follows that the monomials occur in
with equal coefficients.
w
will have a similar cover but with
cover, it follows that each such cover in
isolated node.
Z E(G-x;w)
G
yields a corr-
with respect to
wlm
and
w
Z E(G-x;)
in
nl, j wn2 ,J...wnp’j
is the coefficient of
3
E(G)
It follows that
mials.
Since
can be extended to a cover of
Therefore every monomial m
in
It is therefore the weight
G.
Hence it is a monomial of the polynomial
G.
in
(3. I)
is the weight of a cover with
P
one isolated node less than the corresponding cover in
E(G-x;).
is the number of
p
Then
E(G)
Z
where
Z E(G-x;w)
is
Z
and
E(G-x;)
Hence the result follows.
Throughout the rest of this section, we will assume that the graph has at least
three nodes.
Also ’reconstructible’ would mean node-reconstructible.
By the deck
STAR POLYNOMIALS, GRAPHICAL PARTITIONS AND RECONSTRUCTION
D
G
a graph
G
91
{G-x: xeV(G)}.
we would mean the set
It is well known (see Harary [5], Kelly [6], and Chatrand and Kronk [7]) that
G
is disconnected, and therefore the number
LEMMA 4.
Let
G
graphs with
described.
nodes.
r
graphs with
Suppose that
G
has
the number of isolated nodes in
isolated nodes, G
D
G
impossible unless
D
Since
k > r-l.
G.
r (>0)
isolated
isolated nodes and
G
p-r
G
must be of the form
D
G
G
D
r-I
k _> r.
isolated nodes and only
to form an element of
Clearly
k
k
r.
Therefore
r-I isolated nodes, k#r-l.
r-I
has
G
nodes each of whose removal reduces
themselves isolated nodes.
be
isolated nodes or it would mean that
r-l
elements of
r
k
has at least
is a matching (in which case the result holds).
G
Let
yields at least one new isolated node, and this is
contains no elements with less that
But exactly
r
D
Then
Since every element of
one node can be removed at a time from
has exactly
has
is as described in the theorem.
G
cannot have less than
the removal of each node from
Hence
isolated nodes.
r
Conversely, suppose that
k _> r
G
r-I
or more isolated nodes.
r
PROOF.
p
D
G
is reconstruc-
The following lemma gives a
Then
be a graph with
has exactly
G
DG.
connection between the number of isolated nodes and
nodes if and only if
is reconstructible if
G
of isolated nodes in
We can however, prove the latter independently.
tible.
H
It follows that
disconnected graphs are reconstructible.
r-l.
to
D
G.
G
Therefore
These nodes must be
Therefore
k
r
and the
result follows.
From the above lemma, we see that
It follows that if
br’S
maining
G
can be determined.
Let
G
be a graph with
C(w2,w
,Wp)
Suppose that
D
p
G
is given.
then
Z E(G-x;w)
Then
w
w2,w
can be found.
p
Hence / Z
But this polynomial contains all the simple terms except
Cp_ 2
0, then from Lemma 3,
cients will be defined.
# 0
Wp),
is a polynomial in the weights
We will consider two cases (i)
Cp_ 2
From Theorem 4, we have, by inte-
nodes.
w l,
fore all the simple coefficients of
Cp_ 2
The following result is well known
f(ZE(G-x;w)) + C(w 2,w
E(G)
If
G-
provided that all the re-
is reconstructible.
G
grating both sides with respect to
can be found.
D
We will give different derivation using star polynomials.
H
THEOREM 5.
where
HG
is given, then we can find
(0 < r < n)
(see Tutte [2]).
PROOF.
D
b 0 (of Theorem I) can be obtained from
c
p-i
E(G), except
0
and (ii)
Cp_
0
c
p-I
Cp_ 2
However
There-
P
can be immediately found
# 0.
Therefore all the simple coeffi-
It follows from Theorem I, that
will be unknown.
w
E(G-x;_
b
H
G
can be found.
will be known from
D
G
If
(Lemma 4)
E.J. FARRELL and C.M. DE MATAS
92
will have p-I equations and p-!
Therefore the system of equations in Theorem
It can be solved to find
knowns.
Hence
bp_ I.
b I, b 2,
NG
un-
can be found.
In the proof of Theorem 5, we did not give any useful practical method for finding
H G, when
c
We shall do so now.
0.
p-2
can be written as follows:
The equations of Theorem
+
bp_
p
+
bp_ 2 +
b2
+
+ b
b
(pl )b p-I + (p.2 bp_2 +
2c
c2
=(P2-I )bp-I
+ (p 2 )bp-2 +
p-I
p-2
+ (p_2)bp_2
Cp_2=
(p_2)bp_l
+
+
+
2b2
bl
b2
By adding and subtracting alternate equations, we get
p-2c
+
c
+
c
2
(-l)Pc p-2
Let us denote the L.H.S. of this equation by
S
(-I) p
bp_
S
Since
(-l)Pb p-l"
S.
Then
(3.3)
(S-D0)
can be found from the simple coefficients
can be found (from Lemma 5), b
(3 2)
(-l)Pb p-l"
+
b
+
b
c
r
(r
p-2)
and
can be found from Equation (3.3).
p-2
b’s can be found by using the algorithm suggested by Theorem I.
The following example illustrates the technique.
EXAMPLE 2.
Let
C
G
D
be a graph with
given below in Figure I.
G
G
G
G
O
Figure I.
It can be easily confirmed that
+
2 2
6WlW
2
E(G I)
w
E(G2)
E(G3)
w
E(G)
E(G 5)
w
+
12WlW
+ 3w_22 +
4w
2
4ww + 5WlW 3 + w + w
+ 5w21w2 + 8WlW + 2w22 + 2w
+
2
Therefore
i- E15 E(Gi) 0E(G)w
w + 8ww
5w + 24Wl2W
By integrating with respect to
E(G)
2
+
19ww
2
+
38WlW
+
9w22 +
lOw.
w I, we get
+
10WlW
+
C(w2,w3,w,w5).
p, and
b
0
Hence all the
STAR POLYNOMIALS, GRAPHICAL PARTITIONS AND RECONSTRUCTION
93
Therefore
5 -16
S
D
Since no element of
G
+ 19
10.
19 and c
8, c 2
5, c
p
-2
i0
0.
b
has an isolated node, it follows from Lemma 5 that
From Equation (3.3), we get
b
(-1) 5 (-2-0)
b3
c3
-r4 (r3 )br
b2
c2
r
-rE=3 (2)br
b
2c
2.
4
4
4
3
(2)b3
c2
r
-r__Z2 (1)br
2c
2b 2
2.
8
I0
-4b 4
c
(42)b4
19
4b
3b
6
16
I.
12
2
6
8
0
Hence we get
N
G
(43,32,21).
Suppose that
HG
G
is unigraphic.
is reconstructible from
structible from
D G.
D
G
Then
(Theorem 5).
G
is uniquely determined from
It follows that
G
is
NG"
uniquely
But
recon-
Thus we have the following theorem, which can otherwise be
proved by more elementary methods.
THEOREM 6.
The Reconstruction Conjecture holds for all graphs with unigraphic
partitions.
REFERENCES
I.
2.
3.
4.
5.
6.
7.
FARRELL, E. J., On a Class of Polynomials Associated with the Stars of a Graph
and its Application to Node Disjoint Decompositions of Complete Graphs and
Complete Bipartite Graphs into Stars, Canad. Math. Bull., 22(I)(1978) 35-46.
TUTTE, W. T., "Graph heory", Encycope.dia of Matheatiqs and Its Applications,
Vol. 21 Addison-Wesley, 1984.
FARRELL, E. J., and GRELL, J. C., On Reconstructing the Circuit Polynomial of a
Graph, Caribb. J. Math. 1(3)(1983) 109-119.
FARRELL, E. J., On a General Class of Graph Polynomials, J. Comb. Theory B. 26
(1979) 111-122.
HARARY, F., On the Reconstruction of a Graph from a Collection of Subgraphs,
"T.he.ory of Graphs and.. it Appli.cations"(M. Fiedler ed.) Gordon and Breach,
New York, 1979, 131-146.
KELLY, P. J., On Some Mapping Related to Graphs, Pacific J. Math. 24(1964) 191-19&
CHARTRAND, G. and KRONK, H. V., On Reconstructing Disconnected Graphs, Ann. N. Y.
Acad. Sci. 175(1970), 85-86.