The domination and competition graphs of a tournament

The Domination andCompetition Graphs of aTournament

David C. Fisher,1 J. Richard Lundgren,1*Sarah K. Merz,1† and K. B. Reid2**

1UNIVERSITY OF COLORADO AT DENVER, DENVER, CO 802172CALIFORNIA STATE UNIVERSITY SAN MARCOS, SAN MARCOS, CA 92096

Received December 12, 1994; accepted April 15, 1998

Abstract: Vertices x and y dominate a tournament T if for all vertices z /= x, y,either x beats z or y beats z. Let dom(T ) be the graph on the vertices of T withedges between pairs of vertices that dominate T . We show that dom(T ) is eitheran odd cycle with possible pendant vertices or a forest of caterpillars. While thisis not a characterization, it does lead to considerable information about dom(T ).Since dom(T ) is the complement of the competition graph of the tournamentformed by reversing the arcs of T , complementary results are obtained for thecompetition graph of a tournament. c© 1998 John Wiley & Sons, Inc. J Graph Theory 29: 103–110,

1998

Keywords: tournament, domination, competition, odd cycle, caterpillar

*This research was partially supported by Research Contract N00014-91-J-1145 ofthe Office of Naval Research.† This research was partially supported by Research Contract N00014-93-1-0670of the Office of Naval Research. Author is now at University of Pacific, Stockton,CA 95211.** This research was partially supported by Research Contract N00014-92-J-1400of the Office of Naval Research.

c© 1998 John Wiley & Sons, Inc. CCC 0364-9024/98/020103-08

104 JOURNAL OF GRAPH THEORY

1. INTRODUCTION

Suppose n tennis players compete in a ‘‘round robin’’ tournament, where eachplayer competes against every other player exactly once. If the players have ap-proximately equal abilities and n is large, it is unlikely that one player will beatevery other player. It is more likely that there are two players so that every otherplayer is beaten by at least one of the two (such a pair might form a good ‘‘doubles’’team). We show that regardless of the results, there are at most n such pairs.

The results can be modeled as a ‘‘tournament.’’ A digraph D is a set V (D) ofvertices and a set A(D) of ordered pair of vertices called arcs. We will denote anarc from x to y by (x, y) ∈ A(D) or say that ‘‘x beats y’’ (or ‘‘y loses to x’’). Forall vertices x, letOD(x) orO(x) (the out-set of x) be the subset of the vertices thatx beats. Similarly, let ID(x) or I(x) (the in-set) be the subset of vertices that beat x.Let d+(x) = |O(x)| be the out-degree of x. Let ∆+(D) be the maximum of d+(x)over all vertices x in V (D). A tournament T is a digraph without loops (arcs of theform (x, x)) in which for all x and y (distinct) in V (T ), either (x, y) ∈ A(T ), or(y, x) ∈ A(T ), but not both. An n-tournament is a tournament with n vertices. Aregular tournament is one in which d+(x) is constant for all vertices x. See Moon[10] and Reid and Beineke [12] for more about tournaments.

Given a digraph D, vertices x and y dominate D if O(x) ∪ O(y) ∪ {x, y} =V (D). Let the domination graph of D, denoted dom(D), be the graph on verticesV (D) with edges between the pairs of vertices that dominate D (see Fig. 1).

The domination graph is closely related to the ‘‘competition graph.’’ Given adigraphD, the competition graph ofD, denoted C(D), is the graph on V (D) withan edge between vertices x and y if and only if O(x) ∩ O(y) /= ∅. Competitiongraphs were introduced by Cohen [2, 3] in the study of food webs. A food webcan be modeled by a digraph whose vertices represent various species, with an arcfrom vertex x to vertex y if the species represented by x preys upon the speciesrepresented by y. If two vertices beat a common vertex, this represents two species

FIGURE 1. A tournament and its domination graph. The edges of the domination graphdom(T ) are all pairs of vertices that dominate the tournament T . For example, vertices cand e are adjacent in dom(T ) because in T vertex c beats a, d, f , and g, and vertex e beatsb, c, f , and h.

DOMINATION AND COMPETITION GRAPHS 105

competing for common prey, hence the name ‘‘competition graph.’’ Competitiongraphs and their generalizations have been extensively studied, for example, byBrigham and Dutton [1], Kim, McKee, McMorris, and Roberts [7], and Roberts andRaychaudhuri [11]. Comprehensive surveys on competition graphs are providedby Kim [6] and Lundgren [8].

Competition graphs of tournaments were first considered by Lundgren, Merz,and Rasmussen [9]. Our initial goal was to determine the minimum number ofedges in the competition graph of a tournament with n vertices. We realized thatit is easier to look at the complement of the competition graph as it generally hasfewer edges. The domination graph of a tournament T is the complement of thecompetition graph of the tournament formed by reversing the arcs of T (this is notnecessarily true for digraphs). So for tournaments, results on domination graphscorrespond to results on competition graphs.

2. DOMINATION GRAPHS OF TOURNAMENTS

Which graphs can be the domination graph of a tournament? We partially answerthis question by finding graphs that can never be a subgraph of dom(T ) for anytournament T . An independent set is a subset of the vertices of a graph G with noedges between the vertices in the subset.

Proposition 2.1. Let T be a tournament with z ∈ V (T ). Then O(z) is an inde-pendent set of dom(T ).

Proof. Let x, y ∈ O(z).Then z /∈ O(x) ∪ O(y) ∪ {x, y}. So x and y do notdominate T and, hence, are not adjacent in dom(T ).

Let α(G) (the independence number of G) denote the maximum cardinality ofan independent set of a graph G.

Corollary 2.1. For any tournament T, we have ∆+(T ) ≤ α(dom(T )).Let Cn denote the undirected cycle on n vertices.

Lemma 2.1. Let T be an n-tournament where n ≥ 4 is an even number. ThenCn is not a subgraph of dom(T ).

Proof. Suppose that Cn is a subgraph of dom(T ). Let 1, 2, 3, . . . , n be theconsecutively labeled vertices of Cn. As n is even, ∆+(T ) ≥ n

2 . Corollary 2.1shows ∆+(T ) ≤ α(dom(T )) ≤ α(Cn) = n

2 . So ∆+(T ) = n2 . The only two

possible independent sets of order n2 in dom(T ) are the two maximal independent

sets of Cn given by A = {1, 3, 5, . . . , n − 1} and B = {2, 4, 6, . . . , n}. Withoutloss of generality, assume d+(1) = n

2 . Then by Proposition 2.1, O(1) = B. Thenfor any i ∈ B, we have O(i) /= A, because 1 beats i, and O(i) /= B, becausei ∈ B. For any i ∈ A − {1}, we have O(i) /= A, because i ∈ A, and O(i) /= B,because i beats 1. Thus, d+(i) < n

2 for all i /= 1. So at most one vertex of T canhave out-degree n

2 ; while the other vertices have out-degree n2 − 1 or less. Thus,


the sum of the out-degrees of the vertices of T is at most n2 + (n− 1)(n2 − 1). For

n ≥ 3, this is less than the n(n−1)2 arcs in T , a contradiction.

Let n ≥ 3 be an odd integer. Let S be a (n−12 )-set contained in Zn (the integers

mod n) where 0 /∈ S and s1 +s2 /≡ 0 for all s1, s2 ∈ S. For such sets, we can forma regular tournament T (S) called the rotational tournament with symbol S whosevertices are labeled by the elements of Zn and with arcs (i, j) if j − i ≡ s, wheres ∈ S. In particular, S = {1, 3, 5, . . . , n− 2}, where n is odd satisfies 0 /∈ S ands1 + s2 /≡ 0 for all s1, s2 ∈ S. Thus, we can define Un = T ({1, 3, 5, . . . , n− 2}).Figure 2(a) shows U7 = T ({1, 3, 5}). Figure 2(b) shows T ({1, 2, 4}). The regular7-tournament in Fig. 2(c) is not rotational as O(0) and O(4) induce distinct 3-tournaments.

Lemma 2.2. Let T be an n-tournament where n ≥ 3 is an odd number. ThenCn is a subgraph of dom(T ) if and only if T is isomorphic to Un.

Proof. (⇐) The only dominating pairs in Un are i and j, where j − i ≡ 1 orn − 1. So dom(Un) is the n-cycle with vertices labeled consecutively by 0, 1, 2,. . . , n− 1.

(⇒) Assume that Cn is a subgraph of dom(T ). Consecutively label the ver-tices of Cn by 0, 1, 2, . . . , n − 1. Corollary 2.1 shows ∆+(T ) ≤ α(dom(T )) ≤α(Cn) = n−1

2 . Since the average out-degree in an n-tournament is n−12 , we have

that d+(i) = n−12 for all i. Without loss of generality, assume 0 beats 1. The only

independent set ofCn of order n−12 containing neither 0 nor 1 is {2, 4, 6, . . . , n−1},

so O(1) = {2, 4, 6, . . . , n − 1}. In particular, 1 beats 2. The only independentset of Cn of order n−1

2 containing neither 1 nor 2 is {3, 5, 7, . . . , n}, so O(2) ={3, 5, 7, . . . , n}. Continuing in this way along the vertices of Cn shows that T isisomorphic to Un.

FIGURE 2. Regular tournaments on 7 vertices and their domination graphs.


A tree is a connected acyclic graph. A caterpillar is a tree such that the removalof all pendant vertices (vertices with exactly one neighbor) yields a path. Figure3 shows the smallest tree that is not a caterpillar. This tree will be called NC7(noncaterpillar on 7 vertices). It is well known that a tree that is not a caterpillarmust contain a copy of NC7.

Lemma 2.3. Let T be a 7-tournament. Then NC7 is not a subgraph of dom(T ).

Proof. Suppose NC7 is a subgraph of dom(T ) with the vertices of T labeledas in Fig. 3. Corollary 2.1 shows ∆+(T ) ≤ α(dom(T )) ≤ α(NC7) = 4. Sincethe average out-degree in a tournament with 7 vertices is 3, either T is regular orT has a vertex with out-degree 4. Figure 2 shows the domination graphs of thethree nonisomorphic regular 7-tournaments (it is widely known that these are theonly three nonisomorphic regular 7-tournaments). None of these graphs has NC7as a subgraph. Thus, T has a vertex with out-degree 4. Since {a, c, e, g} is theonly 4 vertex independent set in NC7, by Proposition 2.1 only b, d, or f can haveout-degree 4 in T . Assume without loss of generality that b has out-degree 4. Thenb beats a, c, e, and g, and loses to d and f . Since b and c dominate T , we have cbeats d and f . Now further assume without loss of generality that d beats f . Sincef and g dominate T and c and d both beat f , we see that g beats c and d. But thenc and d do not dominate T , because neither beats g, a contradiction. Thus NC7 isnot a subgraph of the domination graph of a 7-tournament.

Lemma 2.4. Let S be an induced subdigraph of a digraph D. Then the inducedsubgraph of dom(D) on the vertices of S is a subgraph of dom(S).

Proof. Let x, y ∈ S, where {x, y} is an edge in dom(D). ThenOD(x)∪OD(y)∪{x, y} = V (D). As V (S) ⊆ V (D), we haveOS(x)∪OS(y)∪{x, y} = V (S). Thus, {x, y} is an edge in dom(S).

A spiked cycle is a connected graph such that the removal of all pendant verticesyields a cycle.

Theorem 2.1. Let T be an n-tournament. Then dom(T ) is either a spiked oddcycle with or without isolated vertices, or a forest of caterpillars.

Proof. Lemmas 2.1 and 2.4 show that dom(T ) has no even cycles. First assumedom(T ) has a k-cycle C where k is odd. Lemmas 2.2 and 2.4 show that thesubtournament of T on the vertices ofC isUk. By Lemma 2.4, the induced subgraphof dom(T ) on C is a subgraph of dom(Uk) = Ck. So C = Ck is an inducedsubgraph of dom(T ). By Proposition 2.1, if x is not onC, thenO(x) ∩ V (C) is an

FIGURE 3. This graph is not a subgraph of the domination graph of any tournament.


independent set. Since the independent sets in a k-cycle have at most k−12 vertices,

two vertices not in C cannot beat all k vertices in C. So the subgraph induced onthe vertices that are not on C has no edges. If some vertex x that is not on C isadjacent in dom(T ) to at least two vertices onC, say y and z, then edges {x, y} and{x, z} together with one of the two path connecting y and z form an even cycle indom(T ), a contradiction. Thus, dom(T ) is a spiked odd cycle with possibly someisolated vertices.

Otherwise, assume dom(T ) is cycle-free. Then by Lemmas 2.3 and 2.4, eachcomponent must be a caterpillar. So, dom(T ) is a forest of caterpillars.

Proposition 2.2. Any graph G consisting of a spiked odd cycle C with possiblysome isolated vertices is the domination graph of some tournament.

Proof. LetG be such a graph on n vertices with a cycle,C, of length k. Consec-utively label the vertices of C as {0, 1, 2, . . . , k− 1}. For i ∈ {0, 1, 2, . . . , k− 1},let Ni be the set of vertices pendant to i. Let J be the set of isolated vertices.We then construct a tournament T with dom(T ) = G as follows. Create arcs be-tween vertices in V (C) so that the induced subgraph on V (C) is Uk. Let i beatall vertices in Ni. Let all vertices in Ni beat all vertices in V (C) which dominatei, and let all vertices in V (C) that are dominated by i beat all vertices in Ni. Forall i, j ∈ {0, 1, 2, . . . , k − 1} with i /= j, if i beats j, then let each vertex in Ni

beat all vertices in Nj . Let each vertex not in J beat all vertices in J . Pairs ofvertices in Ni and pairs of vertices in J are joined by arcs in an arbitrary manner.The dom(T ) = G.

Not all forests of caterpillars are the domination graph of some tournament (seeFisher, Lundgren, Merz, and Reid [4, 5]). For example, a path with 3 edges is not thedomination graph of any tournament. In a subsequent article, we will examine theproblem of which forests of caterpillars occur as domination graphs of tournaments.

3. CONSEQUENCES OF THE CHARACTERIZATION

Since Theorem 2.1 gives such a detailed description of domination graphs, it isstraightforward to deduce results about various graph parameters for dominationand competition graphs of tournaments. Below are some examples. The bounds inCorollaries 3.1 to 3.6 are achieved for all allowed values of n.

Corollary 3.1. For n ≥ 2, the maximum possible number of edges in the domi-nation graph of an n-tournament is n.

Corollary 3.2. For n ≥ 2, the minimum possible number of edges in the com-petition graph of an n-tournament is (n2 )− n.

A subset of the vertices of G form a clique if there are edges between everypair of vertices in the subset. Let ω(G) (the clique number of G) be the maximumcardinality of a clique ofG. Clearly, α(G) = ω(G), where G is the complement ofG. A coloring ofG is a labeling of its vertices so that adjacent vertices do not have


the same label. Let χ(G) (the chromatic number of G) be the minimum possiblenumber of labels in a coloring of G.

Corollary 3.3. For n ≥ 3, let T be an n-tournament. The clique number andchromatic number of the domination graph of T are at most 3.

A clique cover of G is a labeling of its vertices so that nonadjacent vertices donot have the same label. Let cc(G) (the clique cover number ofG) be the minimumpossible number of labels in a clique cover of G. Clearly, cc(G) = χ(G).

Corollary 3.4. For n ≥ 3, let T be an n-tournament. The independence numberand the clique cover number of the competition graph of T are at most 3.

Corollary 3.5. For n ≥ 3, let T be an n-tournament. The independence numberand the clique cover number of the domination graph of T are at least bn/2c.Corollary 3.6. For n ≥ 3, let T be an n-tournament. The clique number andthe chromatic number of the competition graph of T are at least bn/2c.

A digraph or graph is vertex transitive if for every pair of vertices i and j,there is an automorphism that maps i to j. Rotational tournaments are vertextransitive. Corollary 3.7 shows that the tournamentsUk are the only vertex transitivetournaments that have a dominating pair of vertices. The regular 7-tournamentsshown in Fig. 2 illustrate this result. Figure 2(a) is U7 (which is vertex-transitive)and its domination graph is a 7-cycle. Figure 2(b) is also vertex transitive and itsdomination graph is edgeless. The domination graph of Fig. 2(c) is neither a cyclenor is it edgeless, but this regular tournament is not vertex transitive.

Corollary 3.7. Let T be a vertex-transitive n-tournament. Then either dom(T )is Cn and T is Un, or dom(T ) is edgeless.

Proof. Since a vertex transitive tournament is regular and regular tournamentshave an odd number of vertices, n is odd. Further, since T is vertex transitive,dom(T ) is also vertex transitive. So dom(T ) is a regular graph on an odd numberof vertices. Thus, the common degree of the vertices of dom(T ) is even. Corollary3.1 shows that dom(T ) has at most n edges. Therefore, this degree is either 2 or 0.If it is 2, then dom(T ) is a disjoint union of cycles. However, Theorem 2.1 statesthat dom(T ) can have only one cycle. Thus, dom(T ) is Cn. By Lemma 2.2, wehave that T = Un. Otherwise, the degree is 0 and dom(T ) is edgeless.

References

[1] R. C. Brigham and R. D. Dutton, A characterization of competition graphs,Discrete Applied Math. 6 (1983), 315–317.

[2] J. E. Cohen, Food webs and the dimensionality of trophic niche space, Pro-ceedings of the National Academy of Science, 74 (1977), 4533–4536.

[3] J. E. Cohen and Z. J. Palka, A stochastic theory of community food webs.V.: intervality and triangulation in the trophic-niche overlap graph, AmericanNaturalist 135 (1990), 435–463.


[4] D. C. Fisher, J. R. Lundgren, S. K. Merz, and K. B. Reid, Domination graphsof tournaments and digraphs, Congressus Numeratium 108 (1995) 97–107.

[5] D. C. Fisher, J. R. Lundgren, S. K. Merz, and K. B. Reid, Connected domi-nation graphs of tournaments and digraphs, JCMCC, to appear.

[6] S. Kim, The competition number and its variants. In Quo Vadis, Graph The-ory? J. Gimbel, J. W. Kennedy, and L. V. Quintas, eds. Annals of DiscreteMathematics 55 (1993), 313–326.

[7] S. R. Kim, T. A. McKee, F. R. McMorris, and F. S. Roberts, p-competitiongraphs, Linear Algebra and Its Appl. 217 (1995), 167–178.

[8] J. R. Lundgren, Food webs, competition graphs, competition-common enemygraphs, and niche graphs, in Applications of Combinatorics and Graph Theoryto the Biological and Social Sciences, F. S. Roberts, Ed., Springer–Verlag(1989). In IMH Volumes in Mathematics and Its Applications, 17.

[9] J. R. Lundgren, S. K. Merz, and C. W. Rasmussen, On direct computation ofchromatic numbers of competition graphs, Linear Algebra and Its Appl. 217(1995), 225–249.

[10] J. W. Moon, Topics On Tournaments. Holt, Rinehart and Winston, 1968.[11] A. Raychaudhuri and F. S. Roberts, Generalized competition graphs and their

applications, in Methods of Operations Research, P. Brucker and P. Pauly,Eds., Anton Haim, Konigstein, Germany (1985), 295–311.

[12] K. B. Reid and L. W. Beineke, Tournaments, in Selected Topics in GraphTheory, L. W. Beineke and R. J. Wilson, Eds., Academic Press, New York(1979), 169–204.

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