Competition Graphs of Semiorders
Fred Roberts, Rutgers University
Joint work with Suh-Ryung Kim, Seoul National
1
University
Happy Birthday
Jean-Claude !!
2
Semiorders
The notion of semiorder arose from problems in
utility 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) + 
3
Jean-Claude Falmagne semiorder: Google:
Jean-Claude Falmagne (Chairman)
... Jean-Claude Falmagne (Chairman). ... On the Separation of two Relations
by a Biorder
or Semiorder. ... Falmagne, Mathematical Social Sciences, 11(3), 1987, 1-18.
...
www.highed.aleks.com/corp/jcfbio-ENGLISH.html - 22k - Cached Similar pages
[PDF] The Assessment of Knowledge in Theory and in Practice
File Format: PDF/Adobe Acrobat - View as HTML
... Send correspondence to: Jean-Claude Falmagne, Dept ... We wish to thank
Chris Doble, Dina
Falmagne, and Lin ... a ‘weak order’ or perhaps a ‘semiorder’ (in the ...
www.highed.aleks.com/about/Science_Behind_ALEKS.pdf - Similar pages
UC Irvine Faculty
Jean-Claude Falmagne (949)-824-4880 [email protected]. ... of two
Relations by a Biorder
or Semiorder. ... Falmagne, Mathematical Social Sciences, 11(3), 1987, 1-18.
www.socsci.uci.edu/cogsci/ personnel/falmagne/falmagne.html - 17k - Cached
4
- Similar pages
Jean-Claude Falmagne and Semiorders
•1987 paper with Doignon and Falmagne on
separation of two orders by a biorder or semiorder.
•Work on biorders goes way back – 1969 paper with
Ducamp on composite measurement
5
Jean-Claude Falmagne and Semiorders
•1999 “primer on media theory”
Beginning: “The family of all semiorders on a
finite set has an interesting property: Any
semiorder S can be joined to any other semiorder
S' by successively adding or removing pairs of
elements, without ever leaving the family.”
This observation underlies models of the
evolution of individual preferences under the
influence of the flow of information from the
media
Extensive related work with Doignon,
Regenwetter, Grofman, Ovchinnikov
6
Jean-Claude Falmagne and Semiorders
•A similar idea underlies the notion of well-graded
families of relations (Doignon and Falmagne 1997)
•2001 paper with Doble, Doignon, and Fishburn on
“almost connected orders” – based on one of the
standard axioms for semiorders (no 3-point chain is
incomparable to a fourth point)
•These are just a few examples.
7
Competition Graphs
The notion of competition graph arose from a
problem of ecology.
Key idea: Two species compete if they have a
common prey.
8
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
9
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.
10
bird
fox
insect
grass
deer
bird
insect
fox
deer
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.
12
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.
13
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. 14
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.
15
Structure of Competition Graphs
In studying competition graphs in ecology, Joel
Cohen 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.
16
Interval Graphs
c
a
b
d
e
a
b
e
d
c
17
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.
18
Structure of Competition Graphs
This remarkable empirical observation of
Cohen’s 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: They have no
directed cycles.
19
Structure of Competition Graphs
We are interested in finding out what graphs
are the competition graphs arising from
semiorders.
20
Competition Graphs of Semiorders
Let (V,R) be a semiorder.
Think of it as a digraph with an arc from x to y
if xRy.
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
21
y is sufficiently far to the left of x.
Competition Graphs of Semiorders
The influence application involves a similar
model -- though the linear corridor is a bit farfetched. (We will consider more general
situations soon.)
Note that semiorders are acyclic.
So: What graphs are competition graphs of
semiorders?
22
Graph-Theoretical Notation
Kq is the graph with q vertices and edges
between all of them:
K5
23
Graph-Theoretical Notation
Iq is the graph with q vertices and no edges:
I7
24
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
25
Competition Graphs of Semiorders
So: Is this interesting?
26
Boring!
Really boring!
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.
29
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.
30
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.
31
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.
32
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.
33
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
34
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)?
35
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.36

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
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.
38
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.
39
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).
40
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).
41
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).
42
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.
43
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.
44
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
45
(i). G = K4 – P3  Iq for q > 0
Competition Graphs of Digraphs
Satisfying Condition C(p)
By part (c) of the 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.
46
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. 47
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.
48
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.
49
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
50
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).
51
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.
52
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.
53
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)?
54
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.
55
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
56
Condition C*(p)
It is easy to see that these are all interval graphs.
Question: Can we get a noninterval graph this
way???
57
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.
58
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.
59
Open Problems
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).
61
Closing Remark
62