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Competition Graphs of Semiorders

Fred Roberts, Rutgers UniversityJoint work with Suh-Ryung Kim, Seoul National

University

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Happy Birthday Joel!

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RANDCorporation

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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a

b

c

d

aRb & cRd aRd or cRb

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a

b

c

d

aRb & cRd aRd or cRb

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a

b

c

d

aRb & cRd aRd or cRb

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a

b

c

daRbRc aRd or dRc

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a

b

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daRbRc aRd or dRc

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a

b

c

d

aRbRc aRd or dRc

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Competition GraphsThe 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 WebsFood Webs

Let the vertices of a digraph be species in an ecosystem.

Include an arc from x to y if x preys on y.

fox insect grass deer

bird

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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.

22fox

bird insect

deer grass

fox insect grass deer

bird

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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:CodingChannel assignment in communicationsModeling of complex systems arising from study of energy and economic systemsSpread of opinions/influence in decisionmaking situationsInformation 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

a b d e

c

ab

c

de

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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 pso that we can assign to each vertex of G a box in Euclidean p-spaceso that two vertices are neighborsiff 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 ceiling[N(r,s+1)/2].

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Structure of Competition GraphsThe 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 SemiordersTheorem: 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?

42Boring!

43Really 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:

ab (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

a

b c

d e f

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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 NumberSuppose 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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a b

cd

G = C4

a b c d

α(a,b) α(b,c) α(c,d)

α(a,d)

D

a b

cd

C(D) = G U I4

α(a,b)

α(b,c)

α(c,d)

α(a,d)

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: 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 graphsof 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 (c). G = P3 Iq for q > 0(d). G = P4 Iq for q > 0(e). G = K1,3 Iq for q > 0(f). G = K2 K2 Iq for q > 0(g). G = C4 Iq for q > 1(h). G = K4 – e Iq for q > 0(i). G = K4 – P3 Iq for q > 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 > 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.

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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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DualityGiven 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) > 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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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

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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