Competition Graphs of Semiorders
Fred Roberts, Rutgers University
Joint work with Suh-Ryung Kim, Seoul National
1
University
Happy Birthday Joel!
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RAND
Corporation
Santa Monica,
CA
1968-1971
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Table of Contents:
I. Preference**
II. Scrambling**
k-suitable sets
III. Transitive Subtournaments
IV. Matrices and Line Shifts
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Searching for More Information
about Joel
The results of my Google search
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Semiorders
The notion of semiorder arose from problems in
utility/preference theory and psychophysics
involving thresholds.
V = finite set, R = binary relation on V
(V,R) is a semiorder if there is a real-valued
function f on V and a real number  > 0 so
that for all x, y  V,
(x,y)  R  f(x) > f(y) + 
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Semiorders
Of course, semiorders are special types of partial
orders.
Theorem (Scott and Suppes 1954): A digraph
(with no loops) is a semiorder iff the following
conditions hold:
(1) aRb & cRd  aRd or cRb
(2) aRbRc  aRd or dRc
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aRb & cRd  aRd or cRb
a
c
b
d
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aRb & cRd  aRd or cRb
a
c
b
d
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aRb & cRd  aRd or cRb
a
c
b
d
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a
aRbRc  aRd or dRc
d
b
c
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a
aRbRc  aRd or dRc
b
d
c
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a
b
d
aRbRc  aRd or dRc
c
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Competition Graphs
The notion of competition graph arose from a
problem of ecology.
Key idea: Two species compete if they have a
common prey.
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Competition Graphs of Food Webs
Food Webs
Let the vertices of a digraph be species in
an ecosystem.
Include an arc from x to y if x preys on y.
bird
fox
insect
grass
deer
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Competition Graphs of Food Webs
Consider a corresponding undirected graph.
Vertices = the species in the ecosystem
Edge between a and b if they have a common
prey, i.e., if there is some x so that there are arcs
from a to x and b to x.
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bird
fox
insect
grass
deer
bird
insect
fox
deer
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grass
Competition Graphs
More generally:
Given a digraph D = (V,A).
The competition graph C(D) has vertex set V
and an edge between a and b if there is an x
with (a,x)  A and (b,x)  A.
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Competition Graphs: Other
Applications
Other Applications:
Coding
Channel assignment in communications
Modeling of complex systems arising from
study of energy and economic systems
Spread of opinions/influence in
decisionmaking situations
Information transmission in computer and
communication networks
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Competition Graphs:
Communication Application
Digraph D:
•Vertices are transmitters and
receivers.
•Arc x to y if message sent at x
can be received at y.
Competition graph C(D):
•a and b “compete” if there is a receiver x so
that messages from a and b can both be
received at x.
•In this case, the transmitters a and b interfere. 25
Competition Graphs: Influence
Application
Digraph D:
•Vertices are people
•Arc x to y if opinion of x
influences opinion of y.
Competition graph C(D):
•a and b “compete” if there is a person x so that
opinions from a and b can both influence x.
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Structure of Competition Graphs
In studying competition graphs in ecology, Joel
Cohen (at the RAND Corporation) observed in
1968 that the competition graphs of real food
webs that he had studied were always interval
graphs.
Interval graph: Undirected graph. We can assign
a real interval to each vertex so that x and y are
neighbors in the graph iff their intervals overlap.
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Interval Graphs
c
a
b
d
e
a
b
e
d
c
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Structure of Competition Graphs
Cohen asked if competition graphs of food webs
are always interval graphs.
It is simple to show that purely graphtheoretically, you can get essentially every graph
as a competition graph if a food web can be some
arbitrary directed graph.
It turned out that there are real food webs whose
competition graphs are not interval graphs, but
typically not for “homogeneous” ecosystems.
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Aside: Boxicity and k-Suitable Sets
of Arrangements
More generally, Cohen studied ways to represent
competition graphs as the intersection graphs of
boxes in Euclidean space.
The boxicity of G is the smallest p
so that we can assign to each vertex
of G a box in Euclidean p-space
so that two vertices are neighbors
iff their boxes overlap.
Well-defined but hard to compute.
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Aside: Boxicity and k-Suitable Sets
of Arrangements
A set L of linear orders on a set A of n elements is
called k-suitable if among every k elements a1,
a2, …, ak in A, for every i, there is a linear order
in L in which ai follows all other aj.
N(n,k) = size of smallest k-suitable set L on A.
Notion due to Dushnik who applied it to calculate
dimension of certain partial orders.
Main results about N(n,k) due to Spencer (in his
thesis).
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Aside: Boxicity and k-Suitable Sets
of Arrangements
Let G be a graph and A be a set of q vertices. A
is q-suitable if for every subset B of A with q2 vertices, if a in A-B, there is a vertex x in G
adjacent to all vertices of B and not to a.
Theorem (Cozzens and Roberts 1984): If G
has a 2p-suitable set of vertices, then boxicity
of G is at least p.
Proof uses N(2p,2p-1).
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Aside: Boxicity and k-Suitable Sets
of Arrangements
Let G be a graph and A be a set of r vertices. A
is (r,s)-suitable if for every subset B of A with
s vertices, if a in A-B, there is a vertex x in G
adjacent to all vertices of B and not to a.
Theorem (Cozzens and Roberts 1984): If G
has an (r,s)-suitable set of vertices, then
boxicity of G is at least ceiling[N(r,s+1)/2].
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Structure of Competition Graphs
The remarkable empirical observation of
Cohen’s that real-world competition graphs are
usually interval graphs has led to a great deal of
research on the structure of competition graphs
and on the relation between the structure of
digraphs and their corresponding competition
graphs, with some very useful insights obtained.
Competition graphs of many kinds of digraphs
have been studied.
In many of the applications of interest, the
digraphs studied are acyclic.
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Structure of Competition Graphs
We are interested in finding out what graphs
are the competition graphs arising from
semiorders.
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Competition Graphs of Semiorders
Let (V,R) be a semiorder.
In the communication application: Transmitters
and receivers in a linear corridor and messages
can only be transmitted from right to left.
Because of local interference (“jamming”) a
message sent at x can only be received at y if
y is sufficiently far to the left of x.
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Competition Graphs of Semiorders
In the computer/communication network
application: Think of a hierarchical architecture
for the network.
A computer can only communicate with a
computer that is sufficiently far below it in the
hierarchy.
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Competition Graphs of Semiorders
The influence application involves a similar
model -- the linear corridor is a bit far-fetched,
but the hierarchy model is not.
We will consider more general situations soon.
Note that semiorders are acyclic.
So: What graphs are competition graphs of
semiorders?
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Graph-Theoretical Notation
Iq is the graph with q vertices and no edges:
I7
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Competition Graphs of Semiorders
Theorem: A graph G is the competition graph of
a semiorder iff G = Iq for q > 0 or G = Kr  Iq
for r >1, q > 0.
Proof: straightforward.
K5 U I7
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Competition Graphs of Semiorders
So: Is this interesting?
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Boring!
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Really boring!
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Competition Graphs of Interval
Orders
A similar theorem holds for interval orders.
D = (V,A) is an interval order if there is an
assignment of a (closed) real interval J(x) to
each vertex x in V so that for all x, y  V,
(x,y)  A  J(x) is strictly to the right of J(y).
Semiorders are a special case of interval orders
where every interval has the same length.
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Competition Graphs of Interval
Orders
Interval orders are digraphs without loops
satisfying the first semiorder axiom:
aRb & cRd  aRd or cRb
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Competition Graphs of Interval
Orders
Theorem: A graph G is the competition graph
of an interval order iff G = Iq for q > 0 or G =
Kr  Iq for r >1, q > 0.
Corollary: A graph is the competition graph of
an interval order iff it is the competition graph of
a semiorder.
Note that the competition graphs obtained from
semiorders and interval orders are always interval
graphs.
We are led to generalizations.
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The Weak Order Associated with a
Semiorder
Given a binary relation (V,R), define a new binary
relation (V,) as follows:
ab  (u)[bRu  aRu & uRa  uRb]
It is well known that if (V,R) is a semiorder, then
(V,) is a weak order. This “associated weak order”
plays an important role in the analysis of semiorders.
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The Condition C(p)
We will be interested in a related relation (V,W):
aWb  (u)[bRu  aRu]
Condition C(p), p  2
A digraph D = (V,A) satisfies condition C(p) if
whenever S is a subset of V of p vertices,
there is a vertex x in S so that yWx for all
y  S – {x}.
Such an x is called a foot of set S.
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The Condition C(p)
Condition C(p) does seem to be an interesting
restriction in its own right when it comes to
influence.
It is a strong requirement:
Given any set S of p individuals in a group,
there is an individual x in S so that
whenever x has influence over individual u,
then so do all individuals in S.
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The Condition C(p)
a
b
d
c
e
f
Note that aWc.
If S = {a,b,c}, foot of S is c: we have aWc, bWc
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The Condition C(p)
Claim: A semiorder (V,R) satisfies condition
C(p) for all p  2.
Proof: Let f be a function satisfying:
(x,y)  R  f(x) > f(y) + 
Given subset S of p elements, a foot of S is an
element with lowest f-value. 
A similar result holds for interval orders.
We shall ask: What graphs are competition
graphs of acyclic digraphs that satisfy
condition C(p)?
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Aside: The Competition Number
Suppose D is an acyclic digraph.
Then its competition graph must have an isolated
vertex (a vertex with no neighbors).
Theorem: If G is any graph, adding sufficiently
many isolated vertices produces the competition
graph of some acyclic digraph.
Proof: Construct acyclic digraph D as follows.
Start with all vertices of G. For each edge {x,y}
in G, add a vertex (x,y) and arcs from x and
y to (x,y). Then G together with the isolated
vertices (x,y) is the competition graph of D.52

The Competition Number
b
a
c
b
a
d
D
G = C4
α(a,b)
d
c
a
b
α(b,c)
α(c,d)
α(a,d)
α(a,b)
α(b,c)
C(D) = G U I4
α(c,d)
α(a,d)
d
c
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The Competition Number
If G is any graph, let k be the smallest number
so that G  Ik is a competition graph of some
acyclic digraph.
k = k(G) is well defined.
It is called the competition number of G.
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The Competition Number
Our previous construction shows that
k(C4)  4.
In fact:
• C4  I2 is a competition graph
• C4  I1 is not
• So k(C4) = 2.
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The Competition Number
Competition numbers are known for many
interesting graphs and classes of graphs.
However:
Theorem (Opsut): It is an NP-complete
problem to compute k(G).
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Aside: Opsut’s Conjecture
Let (G) = smallest number of cliques covering
V(G).
N(v) = open neighborhood of v.
Observation: If G is a line graph, then for all
vertices u, (N(u))  2.
Theorem (Opsut, 1982): If G is a line graph,
then k(G)  2, with equality iff for every u,
(N(u)) = 2.
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Aside: Opsut’s Conjecture
Opsut’s Conjecture (1982): Suppose G is any
graph in which (N(u))  2 for all u. Then k(G)
 2, with equality iff for every u, (N(u)) = 2.
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Aside: Opsut’s Conjecture
Hard problem.
Poljak, Wang
Sample Theorem (Wang 1991): Opsut’s
Conjecture holds for all K4-free graphs.
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Back to the Condition C(p)
aWb  (u)[bRu  aRu]
Condition C(p), p  2
A digraph D = (V,A) satisfies condition C(p) if
whenever S is a subset of V of p vertices,
there is a vertex x in S so that yWx for all
y  S – {x}.
Such an x is called a foot of set S.
Question: What are the competition graphs
of digraphs satisfying Condition C(p)?
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Competition Graphs of Digraphs
Satisfying Condition C(p)
Theorem: Suppose that p  2 and G is a
graph. Then G is the competition graph of an
acyclic digraph D satisfying condition C(p) iff
G is one of the following graphs:
(a). Iq for q > 0
(b). Kr  Iq for r > 1, q > 0
(c). L  Iq where L has fewer than p
vertices, q > 0, and q  k(L).
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Competition Graphs of Digraphs
Satisfying Condition C(p)
Note that the earlier results for semiorders and
interval orders now follow since they satisfy
C(2).
Thus, condition (c) has to have L = I1 and
condition (c) reduces to condition (a).
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Competition Graphs of Digraphs
Satisfying Condition C(p)
Corollary: A graph G is the competition graph
of an acyclic digraph satisfying condition C(2)
iff G = Iq for q > 0 or G = Kr  Iq for r >1,
q > 0.
Corollary: A graph G is the competition graph
of an acyclic digraph satisfying condition C(3)
iff G = Iq for q > 0 or G = Kr  Iq for r >1,
q > 0.
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Competition Graphs of Digraphs
Satisfying Condition C(p)
Corollary: Let G be a graph. Then G is the
competition graph of an acyclic digraph
satisfying condition C(4) iff one of the
following holds:
(a). G = Iq for q > 0
(b). G = Kr  Iq for r > 1, q > 0
(c). G = P3  Iq for q > 0, where P3 is the path
of three vertices.
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Competition Graphs of Digraphs
Satisfying Condition C(p)
Corollary: Let G be a graph. Then G is the
competition graph of an acyclic digraph satisfying
condition C(5) iff one of the following holds:
(a). G = Iq for q > 0
(b). G = Kr  Iq for r > 1, q > 0
Kr: r vertices, all edges
(c). G = P3  Iq for q > 0
Pr: path of r vertices
(d). G = P4  Iq for q > 0
Cr: cycle of r vertices
(e). G = K1,3  Iq for q > 0
K1,3: x joined to a,b,c
(f). G = K2  K2  Iq for q > 0
(g). G = C4  Iq for q > 1
K4 – e: Remove one edge
(h). G = K4 – e  Iq for q > 0
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(i). G = K4 – P3  Iq for q > 0
Competition Graphs of Digraphs
Satisfying Condition C(p)
By part (c) of the characterization theorem, the
following are competition graphs of acyclic
digraphs satisfying condition C(p):
L  Iq for L with fewer than p vertices and q >
0, q  k(L).
If Cr is the cycle of r > 3 vertices, then k(Cr) = 2.
Thus, for p > 4, Cp-1  I2 is a competition graph
of an acyclic digraph satisfying C(p).
If p > 4, Cp-1  I2 is not an interval graph.
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Competition Graphs of Digraphs
Satisfying Condition C(p)
Part (c) of the Theorem really says that condition
C(p) does not pin down the graph structure. In
fact, as long as the graph L has fewer than p
vertices, then no matter how complex its
structure, adding sufficiently many isolated
vertices makes L into a competition graph of an
acyclic digraph satisfying C(p).
In terms of the influence and communication
applications, this says that property C(p) really
doesn’t pin down the structure of competition. 67
Duality
Let D = (V,A) be a digraph.
Its converse Dc has the same set of vertices and
an arc from x to y whenever there is an arc
from y to x in D.
Observe: Converse of a semiorder or interval
order is a semiorder or interval order,
respectively.
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Duality
Let D = (V,A) be a digraph.
The common enemy graph of D has the same
vertex set V and an edge between vertices a
and b if there is a vertex x so that there are arcs
from x to a and x to b.
competition graph of D = common enemy graph
of Dc.
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Duality
Given a binary relation (V,R), we will be
interested in the relation (V,W'):
aW'b  (u)[uRa  uRb]
Contrast the relation
aWb  (u)[bRu  aRu]
Condition C'(p), p  2
A digraph D = (V,A) satisfies condition C'(p) if
whenever S is a subset of V of p vertices, there
is a vertex x in S so that xW'y for all
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y  S - {x}.
Duality
By duality:
There is an acyclic digraph D so that G is the
competition graph of D and D satisfies
condition C(p) iff there is an acyclic digraph D'
so that G is the common enemy graph of D'
and D' satisfies condition C'(p).
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Condition C*(p)
A more interesting variant on condition C(p) is
the following:
A digraph D = (V,A) satisfies condition C*(p) if
whenever S is a subset of V of p vertices, there
is a vertex x in S so that xWy for all
y  S - {x}.
Such an x is called a head of S.
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The Condition C*(p)
Condition C*(p) does seem to be an interesting
restriction in its own right when it comes to
influence.
This is a strong requirement:
Given any set S of p individuals in a group,
there is an individual x in S so that
whenever any individual in S has influence
over individual u, then x has influence over
u.
73
The Condition C*(p)
Note: A semiorder (V,R) satisfies condition
C*(p) for all p  2.
Let f be a function satisfying:
(x,y)  R  f(x) > f(y) + 
Given subset S of p elements, a head of S is
an element with highest f-value.
We shall ask: What graphs are competition
graphs of acyclic digraphs that satisfy
condition C*(p)?
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Condition C*(p)
In general, the problem of determining the graphs
that are competition graphs of acyclic digraphs
satisfying condition C*(p) is unsolved.
We know the result for p = 2, 3, 4, or 5.
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Condition C*(p): Sample Result
Theorem: Let G be a graph. Then G is the
competition graph of an acyclic digraph
satisfying condition C*(5) iff one of the
following holds:
(a). G = Iq for q > 0
(b). G = Kr  Iq for r > 1, q > 0
(c). G = Kr - e  I2 for r > 2
(d). G = Kr – P3  I1 for r > 3
(e). G = Kr – K3  I1 for r > 3
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Condition C*(p)
It is easy to see that these are all interval graphs.
Question: Can we get a noninterval graph this
way???
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e
d
c
a
x
b
y
Easy to see that this digraph is acyclic.
C*(7) holds. The only set S of 7 vertices is V.
Easy to see that e is a head of V.
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b
a
x
e
y
d
c
The competition graph has a cycle from a to b
to c to d to a with no other edges among
{a,b,c,d}.
This is impossible in an interval graph.
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Open Problems
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Open Problems
•Characterize graphs G arising as competition
graphs of digraphs satisfying C(p) without
requiring that D be acyclic.
•Characterize graphs G arising as competition
graphs of acyclic digraphs satisfying C*(p).
•Determine what acyclic digraphs satisfying
C(p) or C*(p) have competition graphs that are
interval graphs.
•Determine what acyclic digraphs satisfy
conditions C(p) or C*(p).
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All our best
wishes, Joel
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