Competition Graphs of Semiorders

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Competition Graphs of Semiorders
Fred Roberts, Rutgers University Joint work with
Suh-Ryung Kim, Seoul National University
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Happy Birthday Joel!
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RAND Corporation Santa Monica, CA 1968-1971
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• Table of Contents
• Preference
• II. Scrambling
• k-suitable sets
• III. Transitive Subtournaments
• IV. Matrices and Line Shifts

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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
? gt 0 so that for all x, y ? V, (x,y) ? R ?
f(x) gt 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
• aRb cRd ? aRd or cRb
• (2) aRbRc ? aRd or dRc

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aRb cRd ? aRd or cRb
c
a
d
b
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aRb cRd ? aRd or cRb
a
c
b
d
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aRb cRd ? aRd or cRb
a
c
d
b
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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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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.
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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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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.

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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
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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 graph-theoretically, 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 q-2
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 ceilingN(r,s1)/2.
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Structure of Competition Graphs
The remarkable empirical observation of Cohens
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
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Competition Graphs of Semiorders
Theorem A graph G is the competition graph of
a semiorder iff G Iq for q gt 0 or G Kr ?
Iq for r gt1, q gt 0. Proof straightforward.
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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 gt 0 or G
Kr ? Iq for r gt1, q gt 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)
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) gt 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.
?
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The Competition Number
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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 Opsuts 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 Opsuts Conjecture
Opsuts 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 Opsuts Conjecture
Hard problem. Poljak, Wang Sample Theorem (Wang
1991) Opsuts 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
gt 0 (b). Kr ? Iq for r gt 1, q gt 0 (c). L ? Iq
where L has fewer than p vertices, q gt 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 gt 0 or G Kr ? Iq for r
gt1, q gt 0. Corollary A graph G is the
competition graph of an acyclic digraph
satisfying condition C(3) iff G Iq for q gt
0 or G Kr ? Iq for r gt1, q gt 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 gt 0 (b). G
Kr ? Iq for r gt 1, q gt 0 (c). G P3 ? Iq
for q gt 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 gt 0 (b). G
Kr ? Iq for r gt 1, q gt 0 (c). G P3 ? Iq
for q gt 0 (d). G P4 ? Iq for q gt 0 (e). G
K1,3 ? Iq for q gt 0 (f). G K2 ? K2 ? Iq for
q gt 0 (g). G C4 ? Iq for q gt 1 (h). G K4
e ? Iq for q gt 0 (i). G K4 P3 ? Iq for q
gt 0
Kr r vertices, all edges Pr path of r
vertices Cr cycle of r vertices K1,3 x
joined to a,b,c K4 e Remove one edge
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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 gt 0, q ?
k(L). If Cr is the cycle of r gt 3 vertices,
then k(Cr) 2. Thus, for p gt 4, Cp-1 ? I2
is a competition graph of an acyclic digraph
satisfying C(p). If p gt 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 doesnt pin down the structure of
competition.
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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 y ? S - x.
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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.
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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) gt 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 gt 0 (b). G
Kr ? Iq for r gt 1, q gt 0 (c). G Kr - e ?
I2 for r gt 2 (d). G Kr P3 ? I1 for r gt
3 (e). G Kr K3 ? I1 for r gt 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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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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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