1 Competition Graphs of Semiorders Fred Roberts, Rutgers University Joint work with Suh-Ryung Kim, Seoul National University

2 Happy Birthday Joel!

3 RAND Corporation Santa Monica, CA

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5 Table of Contents: I.Preference** II. Scrambling** k-suitable sets III. Transitive Subtournaments IV. Matrices and Line Shifts

6 Searching for More Information about Joel The results of my Google search

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11 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) + 

12 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

13 a b c d aRb & cRd  aRd or cRb

14 a b c d aRb & cRd  aRd or cRb

15 a b c d aRb & cRd  aRd or cRb

16 a b c d aRbRc  aRd or dRc

17 a b c d aRbRc  aRd or dRc

18 a b c d aRbRc  aRd or dRc

19 Competition Graphs The notion of competition graph arose from a problem of ecology. Key idea: Two species compete if they have a common prey.

20 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. foxinsectgrassdeer bird

21 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.

22 fox bird insect deergrass foxinsectgrassdeer bird

23 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.

24 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

25 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.

26 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.

27 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.

28 Interval Graphs abd e c a b c d e

29 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.

30 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.

31 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 a 1, a 2, …, a k in A, for every i, there is a linear order in L in which a i follows all other a j. 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).

32 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).

33 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].

34 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.

35 Structure of Competition Graphs We are interested in finding out what graphs are the competition graphs arising from semiorders.

36 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.

37 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.

38 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?

39 Graph-Theoretical Notation I q is the graph with q vertices and no edges: I7I7

40 Competition Graphs of Semiorders Theorem: A graph G is the competition graph of a semiorder iff G = I q for q > 0 or G = K r  I q for r >1, q > 0. Proof: straightforward. K 5 U I 7

41 Competition Graphs of Semiorders So: Is this interesting?

42 Boring!

43 Really boring!

44 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.

45 Competition Graphs of Interval Orders Interval orders are digraphs without loops satisfying the first semiorder axiom: aRb & cRd  aRd or cRb

46 Competition Graphs of Interval Orders Theorem: A graph G is the competition graph of an interval order iff G = I q for q > 0 or G = K r  I q 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.

47 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.

48 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.

49 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.

50 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

51 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)?

52 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. 

53 a b c d G = C 4 a b c d α(a,b) α(b,c)α(c,d) α(a,d) D a b c d C(D) = G U I 4 α(a,b) α(b,c) α(c,d) α(a,d) The Competition Number

54 The Competition Number If G is any graph, let k be the smallest number so that G  I k is a competition graph of some acyclic digraph. k = k(G) is well defined. It is called the competition number of G.

55 The Competition Number Our previous construction shows that k(C 4 )  4. In fact: C 4  I 2 is a competition graph C 4  I 1 is not So k(C 4 ) = 2.

56 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).

57 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.

58 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.

59 Aside: Opsut’s Conjecture Hard problem. Poljak, Wang Sample Theorem (Wang 1991): Opsut’s Conjecture holds for all K 4 -free graphs.

60 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)?

61 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). I q for q > 0 (b). K r  I q for r > 1, q > 0 (c). L  I q where L has fewer than p vertices, q > 0, and q  k(L).

62 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 = I 1 and condition (c) reduces to condition (a).

63 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 = I q for q > 0 or G = K r  I q for r >1, q > 0. Corollary: A graph G is the competition graph of an acyclic digraph satisfying condition C(3) iff G = I q for q > 0 or G = K r  I q for r >1, q > 0.

64 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 = I q for q > 0 (b). G = K r  I q for r > 1, q > 0 (c). G = P 3  I q for q > 0, where P 3 is the path of three vertices.

65 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 = I q for q > 0 (b). G = K r  I q for r > 1, q > 0 (c). G = P 3  I q for q > 0 (d). G = P 4  I q for q > 0 (e). G = K 1,3  I q for q > 0 (f). G = K 2  K 2  I q for q > 0 (g). G = C 4  I q for q > 1 (h). G = K 4 – e  I q for q > 0 (i). G = K 4 – P 3  I q for q > 0 K r : r vertices, all edges P r : path of r vertices C r : cycle of r vertices K 1,3 : x joined to a,b,c K 4 – e: Remove one edge

66 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  I q for L with fewer than p vertices and q > 0, q  k(L). If C r is the cycle of r > 3 vertices, then k(C r ) = 2. Thus, for p > 4, C p-1  I 2 is a competition graph of an acyclic digraph satisfying C(p). If p > 4, C p-1  I 2 is not an interval graph.

67 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.

68 Duality Let D = (V,A) be a digraph. Its converse D c 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.

69 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 D c.

70 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}.

71 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).

72 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.

73 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.

74 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)?

75 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.

76 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 = I q for q > 0 (b). G = K r  I q for r > 1, q > 0 (c). G = K r - e  I 2 for r > 2 (d). G = K r – P 3  I 1 for r > 3 (e). G = K r – K 3  I 1 for r > 3

77 Condition C*(p) It is easy to see that these are all interval graphs. Question: Can we get a noninterval graph this way???

78 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. a b c d e x y

79 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. a b c d e x y

80 Open Problems

81 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).

82 All our best wishes, Joel