0
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1
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6 5
1 0
7 2
8 7
9 8
2 9
4 3
3 4
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0 1
4 9 8
6 5 9
1 0 6
3 2 1
7 4 3
8 7 5
9 8 7
5 6 0
0 1 2
2 3 4
7 8
1 7
0 2
6 1
7
8
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0
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9
8
7
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1
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0
1 7
3 8
5 9
1 3 5 2
9 2 4 3
8 9 3 4
7 8 9 5
8 6
2 3
7 0
3 4
6 1
4 5
0 7
5 6
1 9
6 0
2 8
0 1
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7
COM BIN A TOR I A L
MAT HEM A TIC S
YEA R
8 9
9 7
7 8
February 1969 - June 1970
UNIQUELY PARTIALLY ORDERABLE GRAPHS
by
' A '19nert
Martln
Geert Prins
Department of Statistics
University of North Carolina
Chapel Hill, N.C.
Department of Mathematics
Wayne State University
Detroit, Mich.
Institute of Statistics Mimeo Series No. 600.4
FEBRUARY 1969
t Research partially sponsored by the Air Force Office of Scientific Research,
Office of Aerospace Research, United States Air Force under AFOSR Grant No.
68-1406.
UNIQUELY PARTIALLY ORDERABLE GRAPHS
by
Martin Aigner
t
Geert Prins
Department of Statistics
University of North Carolina
Chapel Hill, N.C.
I.
For any binary relation
taking
(x R y) v (y R x).
INTRODUCTION
R on a set
A as the vertex set of
only if
Department of Mathematics
Wayne State University
Detroit, Mich.
G(R)
G(R)
A one may define a graph
and joining two vertices
G(R)
x, y
is undirected, but may have loops.
by
if and
A well-
known type of problem is the characterization of the graphs associated in this
manner with a specific kind of relation.
For example, the graphs of strict
(= irreflexive) partial orders (PO-graphs) have been characterized by Ghoui1a-
Houri [1, 2]
and Gilmore-Hoffman [3], while characterizations of the graphs of
semi-orders (SO-graphs) and indifference systems (I-graphs) were given by
Roberts [5].
All these characterizations are of the Kuratowski-type, i.e.,
they are given in terms of certain sub graphs the graphs under consideration must
1
not contain.
Given a PO-graph
G,
Gilmore and Hoffman also provide an al-
gorithm for obtaining a partial order
R for which
G(R) = G.
In Section II,
we somewhat simplify this algorithm.
If
R is symmetric, then
G(R)
completely determines
not symmetric, then not only do we have
relations
R'
with
G(R')
= G(R).
G(R)
then
R'
Rand
R or
R'
R'
u
= R.
But if
R
is
= G(R) , but there may be other
We shall call a graph
orderable (UPO) or uniquely serni-orderable (USO) if
graph, and if
R.
u 2
G uniquely partially
G is a PO-graph or So-
are two relations such that
G
= G(R) = G(R'),
In Section II, we will investigate in which way di-
rections may be forced by already existing orientations.
Section III shall be
IThese three types of graphs are special cases of the class of perfect graphs,
which, too, have been successfully characterized in the sense of Kuratowski [6].
2~
stands for the inverse relation of
R.
t Research partially sponsored by the Air Force Office of Scientific Research,
Office of Aerospace Research, United States Air Force under AFOSR Grant No.
68-1406.
2
devoted to semi-orders and related structures, and we give a characterization
of USO-graphs there.
In Section IV, we discuss partial orders and present a
sufficient condition for PO-graphs to be uniquely partially orderable.
NOTATION:
1.
Let
G be a graph, then
V(G)
denotes its vertex set, E(G)
its edge-
set.
2.
Let
S
c
duced by
V(G),
S,
then
C
k
4.
n
G between
S.
stands for the cycle of length
plete graph on
and
shall denote the full subgraph or subgraph in-
i.e., the sub graph which contains all edges of
any two points of
3.
~
l
K
m,n
k
(without chords),
K
l
for the com-
for the complete bipartite graph on
m
vertices.
An isolated point will be called trivial component.
In this paper, a graph is assumed to be finite undirected without loops
or multiple edges, all sets considered are finite.
II. STRONG AND TRANSITIVE FORCINGS
Let us begin by giving the Gilmore-Hoffman characterization of PO-graphs.
Definition 1:
A generalized path of length
gression of vertices
aI' a , •.. , a k ,
2
such that no two ordered pairs
k-l
a.
1
in a graph
E
G is a pro-
V(G),
(ai' a + ), (a , a + ) are the same (i.e.,
j l
i l
j
an edge may be traversed twice, but at most once in either direction).
case
a
l
Definition 2:
=~,
we speak of a generalized cycle.
Given the generalized path
aI' " ' ,
~,
an edge (a., a.)
1
J
In
3
with
Ij -
il ~ z
angular chord.
Ij -
is called a chord, if
il
= Z, we speak of a tri-
Similar definitions hold for generalized cycles, modulo
k - 1,
of course.
THEOREM
111.
G is a PO-graph iff
G
contains no generalized cycle of
odd length without triangular chords.
In their proof, Gilmore and Hoffman actually design an algorithm and show
that one arrives at a partial ordering by assigning a direction to an arbitrary
edge, orienting all edges whose direction is determined by previously oriented
edges, choosing again an arbitrary undirected edge, etc.
Since the direction
of an edge may be forced in two possible ways, we give the following definitions.
Definition 3:
(a , a )
l
Z
(c, d)
The generalized path
Definition 4:
The direction of the edge
is said to be strongly forced by the direction of
a l , a Z' ... , ~-l' ~
with
(a, b)
(a, b)
=
if there
(a l , a Z)'
(~-l' ~).
(a
b)
-+
means the edge
The direction of the edge
directions of
(b -+ a), (c
Definition 5:
is called a strong path from
~
if it is without triangular chords.
exists a strong path
(c, d)
a , ... ,
l
-+
(a, b)
(a, b)
(a, c)
and
has the orientation
a -+ b.
is said to be transitively forced by the
(b, c),
i f either
(a
b), (b+ c)
-+
or
b).
G(a, b)
denotes the set of all edges whose direction is forced
by an initial orientation of
(a, b)
and subsequent strong and trans-
itive forcings.
LEMMA 1.
Proof:
If
(c, d)
E
G(a, b),
The hypothesis clearly implies
to verify the inverse inclusion.
If
then
G(c, d)
G(c, d)
(a, b)
E
c
G(a, b).
G(a, b),
G(c, d),
hence we only have
then we are done, so
4
suppose
(a, b)
implies
(c
(c, d)
4 G(c,
d).
+
d),
and assume without loss of generality that
(a
+
b)
Starting the Gilmore-Hoffman algorithm now with the edge
and choosing the direction
(d
+
c)
we may assign
(a
+
b)
by the
nature of the algorithm, thus contradicting the hypothesis.
Lemma 1 gives a partitioning of
E(G)
into orientation-blocks where the
number of blocks represents the degree of freedom we have in constructing our
partial order.
Hence, we have the following
COROLLARY 1.
Let
G
number of partial orders
m
be a PO-graph with
R
with
G(R)
=
G
m
is
orientation-blocks.
m
2 .
G
Then the
is a UPO-graph iff
1.
It is to be noted, however, that some or all of these partial orders may
be isomorphic.
THEOREM 1.
In an orientation-block
G(a, b),
the direction of any edge
strongly forces the direction of every other edge, in other words, transitive
forcing can be replaced by strong forcing.
Proof:
Since strong forcing is easily seen to be a
property, we are able to partition
G(a, b)
symmetric and transitive
into classes
H.
1
with two edges
being in the same class iff their directions force each other strongly.
G(a, b)
= HI +
H
2
+ ... +
H ,
t
we then wish to show
and let us start orienting the edges of
HI'
t
=
1.
Let
Assume the opposite
Among the edges in
G(a, b) - HI
there must be one whose direction is transitively forced by the directions of
two edges in
HI'
Let
(c, e) E H
2
the two corresponding edges
(c-+ d)
and
(d
+
e).
(c, d)
be such an edge and assume w.i.o.g. that
and
(d, e)
in
HI
have been directed
5
We first prove that for any other edge
there exists a pair of edges
(d
-+
g),
(f, d) ,
(f, g)
(d, g)
E
i. e. , the direction of every edge in
directions of a pair of edges in
H
2
E
with directions
H
2
is transitively forced by the
which are joined at
HI
the assertion is correct for edges in
k-l
and let
is a strong path in
(f
-+
g).
(f, g)
H2
with
We may assume
~=
-+
d),
hence the edge
would force
(~-l'
g)
~
(f
-+
E(G),
g),
such that
H
2
=
(aI' a )
2
f,
a k+ l
(d, g)
in
-+
-+
e
_
We proceed by
in
Suppose
H2 ·
•
,
~,
~-l'
(~, ~+l)
e),
~-l)
(d-+ ~-l)
with
HI
=
~+l
(f, g),
and
since otherwise
G,
contrary to the assumption.
(d
d),
-+
then by the induction hypothesis
must exist in
we conclude that
(c, e)
aI' a 2 ,
(c, e), (c
g,
=
d.
(f
that can be reached by a strong path
(~-l' d) , (f, d)
we know there are edges
(f
E
H
2
(f -+ g),
HI
finite induction on the length of a strong path from
of length
with, say,
(f
-+
d)
Finally, since
forces
(d
-+
g),
and the
assertion follows by induction.
Now, suppose we orient
H
2
* G(a, b)
c
G(a, b)
by directing
that there are edges in
H
2
G(a, b) - H
2
H
2
It follows from
whose direction is forced
transitively by the directions of a pair of edges in
plies the existence of two edges in
first.
H .
2
This, however, im-
with a common endpoint, say
transitive orientations, which in turn would imply that
(d, z)
z,
and
is oriented
both ways, a contradiction.
Using Theorem 1, we now obtain the following simplification of the GilmoreHoffman algorithm:
Let
such that
G be a PO-graph.
G(R)
We may find a partial ordering
= G by the following inductive procedure, based
on the number of orientation blocks of
(1)
Choose an arbitrary edge
of all edges in
(aI' b )·
l
R
Orient
G.
(aI' b )
l
of
G and let
E
l
be the set
G which can be reached by a strong path from
(aI' b )
l
and give the other edges of
E
l
the
6
(a , b ).
l
l
direction forced by the initial orientation of
Proceed
to (2).
n-l
(2)
E(G)
If
E.].
L:
i=l
If it is not empty, let
be an arbitrary edge in this set and define
(a , b )
n
is empty, stop.
n
set of all edges of
(a , b )
n
n
G on a strong path from
a direction and assign
A graph
Give
(a , b ).
n
(a , b)<
n
as the
n
n
to the other edges of
direction forced by the orientation of
THEOREM 2.
E
n
E
n
the
Proceed to (2)0
G with at most one non-trivial component and no tri-
angles is a UPO-graph iff it is a PO-graph.
Proof:
Every two edges are on a path, and every path is a strong path.
III.
Definition
~~:
for all
(a)
USa-GRAPHS
The binary relation
x, y,
Z, W E
P
is called a semi-order
on
A iff
A
x P x,
!
(b)
(x P Y
P w)
=-
(x P w v
(c)
(x PyA Y P z)
===;.
(x P w v w P z).
1\
Z
Z
P y),
A semi-order clearly is asymmetric and transitive, hence a (strict) partial
order.
Furthermore, it readily follows from (b) that a SO-graph has at most
one non-trivial component.
As it is
our object to determine the structure of USa-graphs and UPO-graphs,
we note first that a graph
component.
G of either type can have at most one non-trivial
Next, we are going to prove a theorem about the complement
which also applies to both classes of graphs.
C
G
7
THEOREM 3.
nected, then
Proof:
Let
G
for suitable
K
~
=
m,n
C
We first show that
[A.]I s
C
G
i
<
+j.
AI' •. "
G).
Every point of
A.
[A. ]
1.
to
J
two different orderings by directing the remaining edges
[An ]
(b) from
Orient
to
for
It is easy to see
[A.] .
that up to this point we do have a partial order (semi-order).
[A ],
n
An'
is joined to
1.
We orient these edges as follows:
orient the edges from
i ~ n-2,
j,
i
have vertex sets
Assume the
arbitrarily (e.g., the orders induced by, respectively,
1.
A. ,
J
is not con-
m, n.
a partial ordering or semi-ordering of
every point of
G
possesses at most two components.
G
opposite and let the components of
the edges of the
C
G be a UPo- or USa-graph and assume
We now obtain
(a) from
[An_I]
to
Again it is easy to check that both orderings
are well defined.
C
Now let
[AI]
or
have two components with point-sets
G
[A ]
2
Al
and
A ,
If either
2
contains an edge, then we obtain two different orderings by
again orienting the lines of
the remaining lines
[AI]
(a) from
Finally, we remark that
K
m,n
[AI]
and
[A ]
to
[A 2 l,
2
arbitrarily, and then directing
and
(b) from
[A2 ]
to
[AI]'
clearly is both a UPo- and a USa-graph, and the
proof is complete.
THEOREM 4.
phic to
K
m,n
A SO-graph is a USa-graph if and only if it is either isomorfor some
m
and n
or if its complement is connected.
In order to prove Theorem 4, we need the definition of an indifference
system and two theorems due to Roberts, which we will state below as Lemmas 2
and 3.
Definition 7:
system
o
>
0
A binary relation
I
on a set
A is called an indifference
if there exists a real-valued function
such that
function for
1.
x I y
~ If(x) - fey)
I
~
f
o.
on
A and a real number
We call
f
a defining
8
We note that
pletely defines
function
f
LEMMA
G(I)
G
is a SO-graph iff
P
for
I
llil.
(~
such that
Let
E
A
I
I
for
c
such that
f(a)
<
a P b
a, b
E
+a,
b.
feb)
x, yEA
Proof of THEOREM 4:
Let
let P, P'
G ~ K
= m,n
A
~
C
f(a)
G,
<
f
I.
then there exists a defining
feb).
to be equivalent if
g(a)
we have
and
g
A,
and let
(1) a I band
are two defining
G be a SO-graph.
(by Theorem 3).
a, b, c, d
G
such that
As clearly neither
a
tradiction to Lemma 3.
and
b
Hence
<
nor
=
G
=
G(P)
G
g(x) < g(y).
is not connected, then
C
and
G(P')
and
G
+ P,
P'
f, f'
d
<
G
is connected,
U
P'
a P b, a P' b, c P d, d P' c
feb), f'(a)
c
C
If
~
If on the other hand
such that
f(a)
g(b), then for every two non-
<
f(x) < fey)
hence, by Lemma 2, two defining functions
G
defines an indifference system
Then if
and
be two semi-orders with
Then there exist
defined by
com-
and if there exists a pair of non-equivalent elements
equivalent elements
is usa iff
G(I)
be an indifference system on a set
Define
b I c
functions for
C
G
is a semi-order whose graph is
be connected.
(2) a I c
a, b
is clearly symmetric and hence its graph
I.
LEMMA ~lil.
Moreover, if
I
+ P.
and
for the indifference system
f'(b), fCc)
<
fed), f'(d)
<
f'(c).
are equivalent, we obtain a con-
is a USa-graph.
IV. UPO-GRAPHS
We begin this section by quoting a theorem due to Marley.
9
THEOREM
ill.
H
is an I-graph iff
(a)
H
contains no
Kl , 3' s ,
(b)
H
contains no
C ' s,
4
(c)
H
is the complement of a PO-graph.
In
(a), (b)
(and for the remainder of the paper) all subgraphs contained
in a graph are full subgraphs.
This result together with Theorem 4 gives the following
COROLLARY 2.
A sufficient condition for a PO-graph to be uniquely partially
orderable is that its complement be connected and contain neither a
We wish to remove the restriction that
if we do this, we may obtain graphs
C
G
does not contain a
C
4
nor a
C .
4
But
G with more than one non-trivial com-
ponent, and such a graph is clearly not UPo.
The purpose of this section is to
show that if we confine ourselves to graphs with only one non-trivial component
we can indeed omit the
LEMMA 4.
Let
C
condition,
4
be an arbitrary
(b, d»
G,
C
4
G
•
in the complement.
If we delete the edge
G'
(a, c)
Let
(or
is again a PO-graph and contains
0
Proof:
Let us first verify that
in
If in the resulting graph
G.
C
in
then the resulting graph
in its complement
no
K 3's
l,
G be a PO-graph without
[a, b, c, d]
from
We must first prove a number of lemmas,
G'
is a PO-graph.
= G - (a, c)
G'
w.i.o.g. we delete
(a, c)
an odd generalized cycle
C without triangluar chords exists, then it must be of the form
C
= {... a, e, c, •.. }
K
, ,
l3
C'
the edge
(d, e)
for some
E
E(G).
= {... a, e, d, e, c, ... }
e
f
a, b, c, d.
Since
C
G
contains no
But now it is easy to see that
constitutes an odd generalized cycle without
10
triangular chords in
G
(after deletion of possible duplicated edges), con-
tradicting the hypothesis on
G.
Next, suppose the delection of
K ,3 = [c; a, e, f], Le.,
l
c
(a, c)
causes a
is joined to
a, e, f
must be joined in
f
Similarly
in
C
G ,
to
b
to either
C
G
to
must be joined in
e
then
in
C
G
[a, d, e, f]
GCo
hence the set
b
would be a
By the same argument
{a, b, c, d, e, f}
b
or
or
d.
If
C
G .
KI , 3 in
f
G
C
G
in
e
G
e, f
,
suppose
d,
C
in
C
in
{b, d, c, £}
Looking at the subgraph induced by the set
that
K ,3
l
,
with, say,
+a,
b, c, d.
we conclude
w.Lo.g. to
were adjacent to
Thus
d
must be joined
e
cannot be adjacent to
b
in
GC,
induces the full subgraph of fig. I in
a
d.
G.
This graph, however, does not permit
feF-_ _-+_ _---::~e
a partial ordering, in contradiction
to the hypothesis on
G.
fig. I
LEMMA 5.
[a, b, c, d]
Let
G
be a
C
4
be a PO-graph without a
in
C
G
,
and assume
Then the reinsertion of the edge
(a, c)
G'
K 3
l ,
in its complement.
=G-
(a, c)
Let
is a UPO-graph.
does not destroy the UPO-property of
G' •
Proof:
The only way the reinsertion of
unique partial ordering on
e
+ a,
b, c, d,
and, since
could possibly violate the
arises when we have two edges
which are both oriented towc1rd or away from
w.i.o.g. the latter.
G
G'
(a, c)
G'
Since
C
G
does not contain a
is a UPO-graph,
now we conclude that all three edges
orientation-block of
G',
(d, e)
K 3'
I,
must be oriented
(a, e), (c, e), (d, e)
and the lemma follows.
(a, e), (c, e)
e.
Assume
(d, e)
(e
~
exists in
d).
But
are in the same
11
LEMMA 6.
Let
G
be a connected PO-graph whose complement is connected
and does not contain a
G'
=
G - (a, c)
K ,3.
1
G
(a, c)
a
or
exists at least one point
sub graphs
[a, b, c, x]
of the edges
y
(b, x)
containing
a, b, c, d.
~ k
in
C
G
x.
x
and
and
c,
it is joined to both.
and assume
receives
a
We show that this
is connected, there
G
and
c.
Considering the
[a, c, d, x], respectively, we infer the existence
(d, x)
in
G.
Denote by
D(x)
D(x)
the component of
contains none of the points
Suppose we have already verified that all vertices of distance
from
x
are adjacent to all four points
k + 1
at distance
tells us that
y
must be adjacent to
[a, b, c, y]
in
G,
and
k
in
z
(d, y)
Since
GCe
which is adjacent to
to some point
and
C
G
is a UPO-graph.
We wish to show that
be a point of distance
graphs
in
Then, after reinsertion, (a, c)
assumption contradicts the connectivity of
C
C
4
is contained in triangles only, i.e., whenever a vertex
is joined to either
G
be a
According to Theorem 1, the only way the lemma could be false, arises
when the edge
x
[a, b, c, d]
to be a UPO-graph.
a unique orientation, ioe.,
Proof:
Let
C
G
from
x
a
[a, c, d, y]
from
x,
then
C
in
c
y
in
G.
in
G.
Let
is adjacent in
The sub graph
G •
and
a, b, c, d
[a, c, z, y]
Finally, the sub-
establish the existence of
(b, y)
and our lemma follows by induction.
It is worth noting that the following theorem can also be derived by a
direct approach involving a case by case investigation of certain subgraphs.
Since that method is rather lengthy and does not throw light on related concepts, we have adopted the present development
THEOREM 5.
Let
and does not contain a
G
0
be a connected PO-graph whose complement is connected
then
G is uniquely partially orderab1e.
C
G
12
Proof:
We proceed by induction on the number of edges in
ber of points
p-l,
p
fixed throughout.
in which case
The smallest possible number of edges is
is a tree and hence a UPO-graph.
G
G'
theorem holds for all graphs
[E(G)
we are finished by Corollary 2.
in
4
I = p,
[V(G')
with
satisfy the hypotheses of the theorem with
C 's
G, keeping the num-
I
Assume then the
IE(G')
e+l.
I
~ e.
G
Let
If there are no
So let us assume the opposite,
then two possibilities arise:
(a)
Case (a).
(a, c)
[a, b, c, d]
C4
at least one of the edges
(a, c) , (b, d)
There is no such
(b)
=
There exists at least one
Suppose
w.l.o.g.
C
G
in
C
4
(a, c)
Case b.
Let
such that
G.
is not a cut-edge in
is not a cut-edge in
=G-
G'
(a, c)
G.
We delete
satisfies the
Lemmas 5 and 6 now settle this case.
[a, b, c, d]
be a
given the following structure of
edges between
C
G
•
and invoking Lemma 4, we note that
induction hypotheses.
in
A, K, B
are
C
4
C
in
G
Assume
•
w.l.o.g.
that we are
G as displayed in fig. 2, where the only
(a, c)
and
(b, d).
fig. 2
If there are no triangles in
were a triangle in
plus
b (or
a)
A(or
B),
G,
would induce a
KI ,3
in
C
G
If there
then the three points making up the triangle
KI ,3
tains at most one of the two vertices
again induces a
we are finished (Theorem 2).
,
in
C
G
c, d,
•
K con-
Finally, a triangle in
hence the traingle plus
contradicting the hypothesis on
G.
b
or
a
13
COROLLARY 3.
Let
Let every triangle of
C
such that
H
contains no
G
be a PO-graph with at most one non-trivial component.
be contained in a connected full sub graph
K 3.
l ,
Then
G
(x, y)
component there exists a line
with some point of
of
G(x, y).
the direction of
and
G(x, y),
exists a triangle
G
We want to show that
4
G(x, y),
adjacent to
[a, b, c],
G(x, y)
(a, b)
a
or
(a, c)
G(x, y).
E
b,
or
If, on the other hand, every
is adjacent to both, then there
(b, c)
E
G(x, y).
theorem.
(a, c)
H
which is
then clearly the direction of this edge forces
H
By Theorem 5,
G(x, y)
which is incident
pothesis this triangle is contained in a full subgraph
(a, b),
G,
contains at most one non-trivial
If there exists an edge in
a, b,
(a, b),
G.
(a, b), (a, b)
incident with only one of
force
of
is a UPO-graph.
Assume the opposite, then because
point of
H
(i.e., the complement with respect to itself) is connected and
Take an arbitrary line
Proof:
E(G).
G
is a UPO-graph, hence both
But by the hy-
as specified in the
and
(b, c)
and we have arrived at a contradiction.
We conclude with an example of a UPO-graph which is covered by the corollary, but not by the main theorem.
fig. 3
14
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[1]
A. Ghoui1a - Houri,
Caracterisation des graphes nonorientes dont on
peut orienter les aretes de maniere
relation d'ordre,
[2]
A. Ghoui1a - Houri,
a
obtenir 1e graphe d'une
C.R. Acad. Sci. Paris, 254(1962), 1370-1371.
F10ts et tensions dans un graphe,
e
~
Ann. Scient. Ec. Norm. Sup. 3 s. 81(1964), 207-265.
[3]
P. C. Gilmore, A. J. Hoffman,
A characterization of Comparability
Graphs and of Interval Graphs,
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R. D. Luce,
Can. J. Math. 16 (1964), 539-548.
Semiorders and a Theory of Utility Discrimination,
Econometrica 24 (1956), 178-191.
[5]
F. S. Roberts,
Representations of Indifference Relations,
Ph.D. Thesis (1968), Dept. of Math.
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H. Sachs,
Stanford University.
On a Problem of Shannon Concerning Perfect Graphs and Two
More Results on Finite Graphs,
atoria1 Mathematics (1968).
Waterloo Conference on Combin-