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The (1, 2)-step competition graph of a hypertournament

  • Ruijuan Li EMAIL logo , Xiaoting An and Xinhong Zhang
Published/Copyright: June 10, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

In 2011, Factor and Merz [Discrete Appl. Math. 159 (2011), 100–103] defined the ( 1 , 2 ) -step competition graph of a digraph. Given a digraph D = ( V , A ) , the ( 1 , 2 ) -step competition graph of D, denoted C 1 , 2 ( D ) , is a graph on V ( D ) , where x y ∈ E ( C 1 , 2 ( D ) ) if and only if there exists a vertex z ≠ x , y such that either d D − y ( x , z ) = 1 and d D − x ( y , z ) ≤ 2 or d D − x ( y , z ) = 1 and d D − y ( x , z ) ≤ 2 . They also characterized the (1, 2)-step competition graphs of tournaments and extended some results to the ( i , j ) -step competition graphs of tournaments. In this paper, the definition of the (1, 2)-step competition graph of a digraph is generalized to a hypertournament and the (1, 2)-step competition graph of a k-hypertournament is characterized. Also, the results are extended to ( i , j ) -step competition graphs of k-hypertournaments.

Keywords: k-hypertournament; (1, 2)-step competition graph; (i, j)-step competition graph
MSC 2010: 05C65; 05C12; 05C20

1 Terminology and introduction

Let G = ( V , E ) be an undirected graph, or a graph for short. V ( G ) and E ( G ) are the vertex set and the edge set of G, respectively. The complement G c of a graph G is the graph with vertex set V ( G ) in which two vertices are adjacent if and only if they are not adjacent in G. Let G 1 and G 2 be two graphs. The union of G 1 and G 2 , denoted by G 1 ∪ G 2 , is the graph with vertex set V ( G 1 ) ∪ V ( G 2 ) and edge set E ( G 1 ) ∪ E ( G 2 ) .

Let D = ( V , A ) be a directed graph, or a digraph for short. V ( D ) and A ( D ) are the vertex set and the arc set of D, respectively. Let i ≥ 1 , j ≥ 1 . The ( i , j ) -step competition graph of D, denoted C i , j ( D ) , is a graph on V ( D ) , where x y ∈ E ( C i , j ( D ) ) if and only if there exists a vertex z ≠ x , y such that either d D − y ( x , z ) ≤ i and d D − x ( y , z ) ≤ j or d D − x ( y , z ) ≤ i and d D − y ( x , z ) ≤ j . When ( i , j ) = ( 1 , 1 ) , it is also called the competition graph of D.

The notion of competition graph was introduced by Cohen [1] as a means of determining the smallest dimension of ecological phase space. In recent years, many researchers investigated m-step competition graphs of some special digraphs and the competition numbers of some graphs etc. (see [2–4]). Particularly, in 1998, Fisher, Lundgren, Merz and Reid [5] studied the domination graphs and competition graphs of a tournament. Recall that a tournament is an orientation of a complete graph. In 2011, Factor and Merz [6] gave the definition of the ( i , j ) -step competition graph of a digraph. They also characterized the ( 1 , 2 ) -step competition graph of a tournament and extended some results to the ( i , j ) -step competition graph of a tournament. They proved the following theorems related to this paper.

Theorem 1.1

[6] A graph G on n ≥ 5 vertices is the ( 1 , 2 ) -step competition graph of some strong tournament if and only if G is K n , K n − E ( P 2 ) , or K n − E ( P 3 ) .

Theorem 1.2

[6] G, a graph on n vertices, is the ( 1 , 2 ) -step competition graph of some tournament if and only if G is one of the following graphs:

  1. K n , where n ≠ 2 , 3 , 4 ;

  2. K n − 1 ∪ K 1 , where n > 1 ;

  3. K n − E ( P 3 ) , where n > 2 ;

  4. K n − E ( P 2 ) , where n ≠ 1 , 4 , or

  5. K n − E ( K 3 ) , where n ≥ 3 .

Theorem 1.3

[6] If T is a tournament with n vertices, i ≥ 1 and j ≥ 2 , then C i , j ( T ) = C 1 , 2 ( T ) .

Given two integers n and k , n ≥ k > 1 , a k-hypertournament T on n vertices is a pair ( V , A ) , where V is a set of vertices, ∣ V ∣ = n and A is a set of k-tuples of vertices, called arcs, so that for any k-subset S of V , A contains exactly one of the k ! k-tuples whose entries belong to S . As usual, we use V ( T ) and A ( T ) to denote the vertex set and the arc set of T, respectively. Clearly, a 2-hypertournament is merely a tournament. When k = n , the hypertournament has only one arc and it does not have much significance to study. Thus, in what follows, we consider 3 ≤ k ≤ n − 1 .

Let T = ( V , A ) be a k-hypertournament on n vertices. For an arc a of T, T − a denotes a hyperdigraph obtained from T by removing the arc a and a ¯ denotes the set of vertices contained in a . If v i , v j ∈ a ¯ and v i precedes v j in a , we say that v i dominates v j in a . We also say the vertex v j is an out-neighbour of v i and use the following notation:

N T + ( v i ) = { v j ∈ V ⧹ { v i } : v i precedes v j in some arc a ∈ A ( T ) } .

We will omit the subscript T if the k-hypertournament T is known from the context.

A path P in a k-hypertournament T is a sequence v 1 a 1 v 2 a 2 v 3 ⋯ v t − 1 a t − 1 v t of distinct vertices v 1 , v 2 , … , v t , t ≥ 1 and distinct arcs a 1 , a 2 , … , a t − 1 such that v i precedes v i + 1 in a i , 1 ≤ i ≤ t − 1 . Meanwhile, let the vertex set V ( P ) = { v 1 , v 2 , … , v t } and the arc set A ( P ) = { a 1 , a 2 , … , a t − 1 } . The length of a path P is the number of its arcs, denoted ℓ ( P ) . A path from x to y is an ( x , y ) -path. The k-hypertournament T is called strong if T has an ( x , y ) -path for every pair x , y of distinct vertices in T.

A k-hypertournament T is said to be transitive if its vertices are labeled v 1 , v 2 , … , v n in such an order so that i < j if and only if v i precedes v j in each arc containing v i and v j .

Now we generalize the ( 1 , 2 ) -step competition graph of a digraph to a k-hypertournament.

Definition 1.4

The ( i , j ) -step competition graph of a k-hypertournament T with i ≥ 1 and j ≥ 1 , denoted C i , j ( T ) , is a graph on V ( T ) , where x y ∈ E ( C i , j ( T ) ) if and only if there exist a vertex z ≠ x , y and an ( x , z ) -path P and a ( y , z ) -path Q satisfying the following:

  1. y ∉ V ( P ) , x ∉ V ( Q ) ;

  2. ℓ ( P ) ≤ i and ℓ ( Q ) ≤ j , or ℓ ( Q ) ≤ i and ℓ ( P ) ≤ j ;

  3. P and Q are arc-disjoint.

If x y ∈ E ( C i , j ( T ) ) , we say that x and y ( i , j ) -step compete. In particular, we say that x and y compete if ℓ ( P ) = 1 and ℓ ( Q ) = 1 . C 1 , 1 ( T ) is also called the competition graph of the k-hypertournament T. Clearly, when k = 2 , T is a tournament and C i , j ( T ) is the ( i , j ) -step competition graph of T.

The k-hypertournaments form one of the most interesting class of digraphs. For the class of k-hypertournaments, the popular topics are the Hamiltonicity and vertex-pancyclicity (see [7–10]). Besides, some researchers investigated the degree sequences and score sequences of k-hypertournaments (see [11,12]). Recently, the H -force set of a hypertournament was also studied (see [13]). In this paper, we study the ( 1 , 2 ) -step competition graph of a k-hypertournament and extend Theorems 1.1–1.3 to k-hypertournaments.

In Sections 2 and 3, useful lemmas are provided to make the proof of the main results easier. In Sections 4 and 5, the ( 1 , 2 ) -step competition graph of a (strong) k-hypertournament is characterized. In Section 6, the main results are extended to the ( i , j ) -step competition graph of a k-hypertournament.

2 The missing edges of C 1 , 2 ( T )

Let T = ( V , A ) be a k-hypertournament. For a pair of distinct vertices x and y in T, A T ( x , y ) denotes the set of all arcs of T in which x precedes y , A T { x , y } denotes the set of all arcs containing x , y in T and A T ∗ { x , y } denotes the set of all arcs containing x , y in T and in which neither x nor y is the last entry.

Lemma 2.1

Let T be a k-hypertournament with n vertices, where 3 ≤ k ≤ n − 1 . Then x y ∉ E ( C 1 , 2 ( T ) ) if and only if one of the following holds:

  1. N + ( x ) = ∅ ;

  2. N + ( y ) = ∅ ;

  3. N + ( x ) = { y } ;

  4. N + ( y ) = { x } ;

  5. A T ∗ { x , y } contains exactly an arc a , and N T − a + ( x ) ⊆ { y } , N T − a + ( y ) ⊆ { x } .

Proof

First, we show the “if” part. Clearly, if one of (a)–(d) holds, we have x y ∉ E ( C 1 , 2 ( T ) ) . Now we assume that the argument (e) holds. Since A T ∗ { x , y } contains exactly an arc a , and N T − a + ( x ) ⊆ { y } , N T − a + ( y ) ⊆ { x } , we have to use the unique arc a to obtain the out-neighbour except y of x and the out-neighbour except x of y . So x and y are impossible to ( 1 , 2 ) -step compete and hence x y ∉ E ( C 1 , 2 ( T ) ) .

Now we show the “only if” part. Assume that x y ∉ E ( C 1 , 2 ( T ) ) . Also, assume that x and y do not satisfy (a)–(d). That means N + ( x ) ⧹ { y } ≠ ∅ , N + ( y ) ⧹ { x } ≠ ∅ . Now we show that x and y satisfy ( e ) . Suppose A T ∗ { x , y } consists of at least two arcs, say a 1 , a 2 ∈ A T ∗ { x , y } . Let w i be the last entry of a i for i = 1 , 2 . If w 1 = w 2 , then x and y compete, a contradiction. So assume w 1 ≠ w 2 . Note that n − 2 k − 2 − 2 ≤ ∣ A T { w 1 , w 2 } ⧹ { a 1 , a 2 } ∣ ≤ n − 2 k − 2 . For ( n , k ) ≠ ( 4 , 3 ) , we have ∣ A T { w 1 , w 2 } ⧹ { a 1 , a 2 } ∣ ≥ 1 . For ( n , k ) = ( 4 , 3 ) , since both a 1 and a 2 contain x , y and a 1 ≠ a 2 , we have a 1 , a 2 ∉ A T { w 1 , w 2 } and hence ∣ A T { w 1 , w 2 } ⧹ { a 1 , a 2 } ∣ = ∣ A T { w 1 , w 2 } ∣ = 2 . Let a 3 ∈ A T { w 1 , w 2 } ⧹ { a 1 , a 2 } . W.l.o.g., a 3 ∈ A T ( w 1 , w 2 ) . Then P = x a 1 w 1 a 3 w 2 and Q = y a 2 w 2 are the paths such that x and y ( 1 , 2 )-step compete, a contradiction. So A T ∗ { x , y } consists of exactly an arc a . Suppose N T − a + ( x ) ⧹ { y } ≠ ∅ . Then there exists an arc b distinct from a such that x has an out-neighbour distinct from y in b . Similar to the aforementioned proof, whether or not the last entries of a and b are same, we always have x y ∈ E ( C 1 , 2 ( T ) ) , a contradiction. So N T − a + ( x ) ⊆ { y } . Similarly, N T − a + ( y ) ⊆ { x } . Thus, A T ∗ { x , y } contains exactly an arc a , and N T − a + ( x ) ⊆ { y } , N T − a + ( y ) ⊆ { x } .

The lemma holds.□

By the proof of Lemma 2.1, we obtain the following result.

Corollary 2.2

Let T be a strong k-hypertournament with n vertices, where 3 ≤ k ≤ n − 1 . Then x y ∉ E ( C 1 , 2 ( T ) ) if and only if one of the following holds:

  1. N + ( x ) = { y } ;

  2. N + ( y ) = { x } ;

  3. A T ∗ { x , y } contains exactly an arc a , and N T − a + ( x ) ⊆ { y } , N T − a + ( y ) ⊆ { x } .

3 The forbidden subgraphs of ( C 1 , 2 ( T ) ) c

Lemma 3.1

Let G on n vertices be the ( 1 , 2 ) -step competition graph of some k-hypertournament T, where 3 ≤ k ≤ n − 1 . Then the complement G c of G does not contain a pair of disjoint edges.

Proof

Suppose the complement G c of G contains a pair of disjoint edges, say x y and z w . So x y , z w ∉ E ( G ) and x , y , z , w are distinct. By Lemma 2.1, we have x y satisfies one of the cases (a)–(e) and z w satisfies one of the cases (a)–(e).

Suppose that at least one of x y and z w satisfies one of the cases (a)–(d). W.l.o.g., we assume that N + ( x ) ⊆ { y } . Then it must be true that the vertex z dominates x in each arc containing x , z but not containing w . Meanwhile, it must be true that the vertex w dominates x in each arc containing x , w but not containing z . So z and w compete and hence z w ∈ E ( C 1 , 2 ( T ) ) = E ( G ) , a contradiction. Thus, both x y and z w satisfy ( e ) .

However, since A T ∗ { x , y } contains exactly an arc a , and N T − a + ( x ) ⊆ { y } , N T − a + ( y ) ⊆ { x } , we have the vertex x must be the last entry in each arc containing x , z , w but not containing y and the vertex y must be the last entry in each arc containing y , z , w but not containing x . Thus, A T ∗ { z , w } contains at least two arcs, which contradicts the fact that z w satisfies ( e ) .

The lemma holds.□

Lemma 3.2

Let G on n vertices be the ( 1 , 2 ) -step competition graph of some k-hypertournament T, where 3 ≤ k ≤ n − 1 . Then the complement G c of G does not contain 3-cycle.

Proof

Suppose to the contrary that the complement G c of G contains 3-cycle, say x y z x . So x y , x z , y z ∉ E ( G ) and x , y , z are distinct. By Lemma 2.1, we have x y , x z and y z satisfy one of the cases (a)–(e), respectively.

Claim 1

None of x y , x z and y z satisfies the case ( a ) or ( b ) .

Proof

Suppose at least one of x y , x z and y z satisfies the case ( a ) or ( b ) . W.l.o.g., we assume that x y satisfies ( a ) , i.e. N + ( x ) = ∅ . Then it must be true that the vertex y dominates x in each arc containing x , y but not containing z . Also, the vertex z dominates x in each arc containing x , z but not containing y . So y and z compete and y z ∈ E ( C 1 , 2 ( T ) ) = E ( G ) , a contradiction. Thus, none of x y , x z and y z satisfies the case ( a ) or ( b ) .□

Claim 2

At most one of x y , x z and y z satisfies the case ( c ) or ( d ) .

Proof

Suppose at least two edges among x y , x z and y z satisfy the case ( c ) or ( d ) . W.l.o.g., we assume that both x y and y z satisfy ( c ) or ( d ) . We consider the following four cases.

Case 1: Both x y and y z satisfy ( c ) . It means that N + ( x ) = { y } , N + ( y ) = { z } . If x z satisfies ( c ) , i.e. N + ( x ) = { z } , it contradicts N + ( x ) = { y } . If x z satisfies ( d ) , i.e. N + ( z ) = { x } , the arcs containing simultaneously x , y , z do not satisfy N + ( x ) = { y } , N + ( y ) = { z } and N + ( z ) = { x } , a contradiction. If x z satisfies ( e ) , i.e. A T ∗ { x , z } contains exactly an arc a , then there exists a vertex w such that w ∈ N + ( x ) . Since N + ( x ) = { y } , we have w = y . So x is the second last entry, y is the last entry and z is any other entry in a . Then the vertex z dominates y in a . Also, the vertex x dominates y in each arc containing x , y but not containing z . So x z ∈ E ( C 1 , 2 ( T ) ) = E ( G ) , a contradiction.

Case 2: Both x y and y z satisfy ( d ) . It means that N + ( y ) = { x } , N + ( z ) = { y } . Similarly to Case 1, we can also get a contradiction.

Case 3: x y satisfies ( c ) and y z satisfies ( d ) . It means that N + ( x ) = { y } , N + ( z ) = { y } . Now the arcs containing simultaneously x , y , z do not satisfy N + ( x ) = { y } , N + ( z ) = { y } , a contradiction.

Case 4: x y satisfies ( d ) and y z satisfies ( c ) . It means that N + ( y ) = { x } and N + ( y ) = { z } . Then x = z , a contradiction.

Thus, at most one of x y , x z and y z satisfies the case ( c ) or ( d ) .□

Claim 3

At most one of x y , x z and y z satisfies the case ( e ) .

Proof

Suppose at least two edges among x y , x z and y z satisfy the case ( e ) . W.l.o.g., we assume that both x z and y z satisfy ( e ) . From the assumption that x z satisfies ( e ) , we get the vertex y dominates x in each arc containing x , y but not containing z . From the assumption that y z satisfies ( e ) , we get the vertex x dominates y in each arc containing x , y but not containing z . This is a contradiction. Thus, at most one of x y , x z and y z satisfies the case ( e ) .□

By Claims 1–3, it is impossible that x y , x z , y z ∉ E ( G ) hold simultaneously. Thus, the complement G c of G does not contain 3-cycle. The lemma holds.□

Lemma 3.3

Let G on n vertices be the ( 1 , 2 ) -step competition graph of some k-hypertournament T, where 3 ≤ k ≤ n − 1 . Then the complement G c of G does not contain K 1 , 3 , unless G = K n − 1 ∪ K 1 .

Proof

Let T be a k-hypertournament on n vertices, where 3 ≤ k ≤ n − 1 , and G the ( 1 , 2 ) -step competition graph of T. Assume G ≠ K n − 1 ∪ K 1 . Now we show that the complement G c of G does not contain K 1 , 3 . Suppose not. Let { x , y , z , w } and { x y , x z , x w } be the vertex set and edge set of the subgraph K 1 , 3 , respectively. So x y , x z , x w ∉ E ( G ) . By Lemma 2.1, we have x y , x z and x w satisfy one of the cases (a)–(e).

Claim 1

None of x y , x z and x w satisfies the case ( a ) or ( b ) .

Proof

Suppose at least one of x y , x z and x w satisfies the case ( a ) or ( b ) . W.l.o.g., we assume that x y satisfies ( a ) , i.e. N + ( x ) = ∅ . Let V ( T ) = { v 1 , v 2 , … , v n } and x = v n . By Lemma 2.1, for all 1 ≤ i ≤ n − 1 , we have v i v n ∉ E ( C 1 , 2 ( T ) ) . We claim that for all 1 ≤ i < j ≤ n − 1 , v i v j ∈ E ( C 1 , 2 ( T ) ) . Indeed,

  • For 1 ≤ i < j ≤ n − ( k − 1 ) , the vertex v i dominates v n by the arc consisting of v i , … , v i + ( k − 3 ) , v n − 1 , v n and the vertex v j dominates v n by the arc consisting of v j , … , v j + ( k − 3 ) , v n − 1 , v n . Then v i and v j compete and v i v j ∈ E ( C 1 , 2 ( T ) ) .

  • For n − ( k − 2 ) ≤ i < j ≤ n − 1 , the vertex v i dominates v n by the arc consisting of v n − k , … , v n − 3 , v n − 1 , v n for n − ( k − 2 ) ≤ i ≤ n − 3 and by the arc consisting of v 1 , … , v 1 + ( k − 3 ) , v n − 2 , v n for i = n − 2 and the vertex v j dominates v n by the arc consisting of v n − ( k − 1 ) , … , v n − 2 , v n − 1 , v n . Then v i and v j compete and v i v j ∈ E ( C 1 , 2 ( T ) ) .

  • For 1 ≤ i ≤ n − k and n − ( k − 2 ) ≤ j ≤ n − 1 , the vertex v i dominates v n by the arc consisting of v i , … , v i + ( k − 3 ) , v n − 1 , v n and the vertex v j dominates v n by the arc consisting of v n − ( k − 1 ) , … , v n − 2 , v n − 1 , v n . Then v i and v j compete and v i v j ∈ E ( C 1 , 2 ( T ) ) .

  • For i = n − ( k − 1 ) and n − ( k − 2 ) ≤ j ≤ n − 1 , the vertex v n − ( k − 1 ) dominates v n by the arc consisting of v 1 , … , v 1 + ( k − 3 ) , v n − 2 , v n for k = 3 and by the arc consisting of v n − k , v n − ( k − 1 ) , … , v n − 3 , v n − 1 , v n for 4 ≤ k ≤ n − 1 and the vertex v j dominates v n by the arc consisting of v n − ( k − 1 ) , … , v n − 2 , v n − 1 , v n . Then v j and v n − ( k − 1 ) compete and v j v n − ( k − 1 ) ∈ E ( C 1 , 2 ( T ) ) .

Then C 1 , 2 ( T ) = K n − 1 ∪ K 1 , a contradiction. Thus, none of x y , x z and x w satisfies the case ( a ) or ( b ) .□

Claim 2

At most one of x y , x z and x w satisfies the case ( c ) or ( d ) .

Proof

Suppose at least two edges among x y , x z and x w satisfy the case ( c ) or ( d ) . W.l.o.g., we assume that both x y and x z satisfy the case ( c ) or ( d ) . We consider the following four cases.

Case 1: Both x y and x z satisfy ( c ) . It means that N + ( x ) = { y } and N + ( x ) = { z } . Then y = z , a contradiction.

Case 2: Both x y and x z satisfy ( d ) . It means that N + ( y ) = { x } and N + ( z ) = { x } . Now the arcs containing simultaneously x , y , z do not satisfy N + ( y ) = { x } , N + ( z ) = { x } , a contradiction.

Case 3: x y satisfies ( c ) and x z satisfies ( d ) . It means that N + ( x ) = { y } and N + ( z ) = { x } . If x w satisfies ( c ) , then N + ( x ) = { w } , contradicting N + ( x ) = { y } . If x w satisfies ( d ) , then N + ( w ) = { x } . Now the arcs containing simultaneously x , z , w do not satisfy N + ( z ) = { x } , N + ( w ) = { x } , a contradiction. If x w satisfies ( e ) . Let b be an arc containing x , z , w . N + ( x ) = { y } yields z dominates x in b . N + ( z ) = { x } yields z is the second last entry, x is the last entry and w is any other entry of b . Clearly, b ∉ A T ∗ { x , w } , i.e. z ∈ N T − a + ( w ) , which contradicts the fact that N T − a + ( w ) ⊆ { x } .

Case 4: x y satisfies ( d ) and x z satisfies ( c ) . Similarly to Case 3, we can also get a contradiction.

Thus, at most one of x y , x z and x w satisfies the case ( c ) or ( d ) .□

Claim 3

At most one of x y , x z and x w satisfies the case ( e ) .

Proof

Suppose at least two edges among x y , x z and x w satisfy the case ( e ) . Assume that x y and x z satisfy ( e ) . From the assumption that x y satisfies ( e ) , we get the vertex z dominates y in each arc containing y , z but not containing x . From the assumption that x z satisfies ( e ) , we get the vertex y dominates z in each arc containing y , z but not containing x . This is a contradiction. Thus, at most one of x y , x z and x w satisfies the case ( e ) .□

By Claims 1–3, it is impossible that x y , x z , x w ∉ E ( G ) hold simultaneously. Thus, the complement G c of G does not contain K 1 , 3 unless G = K n − 1 ∪ K 1 . The lemma holds.□

By Corollary 2.2 and the proof of Lemma 3.3, we obtain the following result.

Corollary 3.4

Let G on n vertices be the ( 1 , 2 ) -step competition graph of some strong k-hypertournament T, where 3 ≤ k ≤ n − 1 . Then the complement G c of G does not contain K 1 , 3 .

4 Strong k-hypertournaments

Theorem 4.1

A graph G on n vertices is the ( 1 , 2 ) -step competition graph of some strong k-hypertournament T with 3 ≤ k ≤ n − 1 if and only if G is K n , K n − E ( P 2 ) , or K n − E ( P 3 ) .

Proof

We first show the “if” part. Let T be a transitive k-hypertournament with the vertices v 1 , v 2 , … , v n . Let T 1 be a k-hypertournament obtained from T by replacing the arc ( v 1 , v 2 , v n − ( k − 3 ) , … , v n ) with ( v n , … , v n − ( k − 3 ) , v 2 , v 1 ) . It is easy to check that T 1 is strong. Now we show that C 1 , 2 ( T 1 ) = K n − E ( P 2 ) . For convenience, let a = ( v n , … , v n − ( k − 3 ) , v 2 , v 1 ) . We claim that v n − 1 v n ∉ E ( C 1 , 2 ( T 1 ) ) . Indeed, for k = 3 , the vertex v n − 1 has a unique out-neighbour v n and Corollary 2.2 ( a ) implies v n − 1 v n ∉ E ( C 1 , 2 ( T 1 ) ) . For 4 ≤ k ≤ n − 1 , A T 1 ∗ { v n − 1 , v n } contains exactly an arc a , and N T 1 − a + ( v n − 1 ) = { v n } , N T 1 − a + ( v n ) = ∅ . Corollary 2.2 ( c ) implies v n − 1 v n ∉ E ( C 1 , 2 ( T 1 ) ) . We also claim v i v j ∈ E ( C 1 , 2 ( T 1 ) ) for all { i , j } ≠ { n − 1 , n } . W.l.o.g., we assume i < j .

  • For 1 ≤ i < j ≤ n − ( k − 1 ) , v i dominates v n by the arc ( v i , v n − ( k − 2 ) , … , v n − 1 , v n ) and v j dominates v n by the arc ( v j , v n − ( k − 2 ) , … , v n − 1 , v n ) . Then v i and v j compete and v i v j ∈ E ( C 1 , 2 ( T 1 ) ) .

  • For n − ( k − 2 ) ≤ i < j ≤ n − 1 , v i dominates v n by the arc ( v 1 , v n − ( k − 2 ) , … , v i , … , v j , … , v n − 1 , v n ) and v j dominates v n by the arc ( v 2 , v n − ( k − 2 ) , … , v i , … , v j , … , v n − 1 , v n ) . Then v i and v j compete and v i v j ∈ E ( C 1 , 2 ( T 1 ) ) .

  • For i = 1 and n − ( k − 2 ) ≤ j ≤ n − 1 , v 1 dominates v n by the arc ( v 1 , v n − ( k − 2 ) , … , v n − 1 , v n ) and v j dominates v n by the arc ( v 2 , v n − ( k − 2 ) , … , v j , … , v n − 1 , v n ) . Then v 1 and v j compete and v 1 v j ∈ E ( C 1 , 2 ( T 1 ) ) .

  • For 2 ≤ i ≤ n − ( k − 1 ) and n − ( k − 2 ) ≤ j ≤ n − 1 , v i dominates v n by the arc ( v i , v n − ( k − 2 ) , … , v n − 1 , v n ) and v j dominates v n by the arc ( v 1 , v n − ( k − 2 ) , … , v j , … , v n − 1 , v n ) . Then v i and v j compete and v i v j ∈ E ( C 1 , 2 ( T 1 ) ) .

  • For i = 1 and j = n , v n dominates v 2 by the arc a , v 2 dominates v n − 1 by the arc ( v 2 , v n − ( k − 2 ) , … , v n − 1 , v n ) and v 1 dominates v n − 1 by the arc ( v 1 , v n − ( k − 2 ) , … , v n − 1 , v n ) . Then v 1 and v n ( 1 , 2 )-step compete and v 1 v n ∈ E ( C 1 , 2 ( T 1 ) ) .

  • For 2 ≤ i ≤ n − 2 and j = n , v n dominates v 1 by the arc a , v 1 dominates v n − 1 by the arc ( v 1 , v n − ( k − 2 ) , … , v n − 1 , v n ) and v i dominates v n − 1 by ( v i , v n − ( k − 2 ) , … , v n − 1 , v n ) for 2 ≤ i ≤ n − ( k − 1 ) and by ( v 2 , v n − ( k − 2 ) , … , v i , … , v n − 1 , v n ) for n − ( k − 2 ) ≤ i ≤ n − 2 . Then v i and v n ( 1 , 2 )-step compete and v i v n ∈ E ( C 1 , 2 ( T 1 ) ) .

Thus, C 1 , 2 ( T 1 ) = K n − E ( P 2 ) .

Let T 2 be a k-hypertournament obtained from T 1 above by replacing the arc ( v 1 , v 2 , v n − ( k − 2 ) , … , v n − 1 ) with ( v n − 1 , … , v n − ( k − 2 ) , v 2 , v 1 ) . It is easy to check that T 2 is strong. Now we show that C 1 , 2 ( T 2 ) = K n .

  • For { i , j } ≠ { n − 1 , n } , similarly to the proof of T 1 , we have v i v j ∈ E ( C 1 , 2 ( T 2 ) ) .

  • For i = n − 1 and j = n , v n − 1 dominates v 1 by the arc ( v n − 1 , … , v n − ( k − 2 ) , v 2 , v 1 ) and v n dominates v 1 by the arc ( v n , … , v n − ( k − 3 ) , v 2 , v 1 ) . Then v n − 1 and v n compete and v n − 1 v n ∈ E ( C 1 , 2 ( T 2 ) ) .

Thus, C 1 , 2 ( T 2 ) = K n .

Let T 3 be a k-hypertournament with the vertices v 1 , v 2 , … , v n satisfying the following:

  1. Each arc excluding v 1 , v 2 satisfies i < j if and only if v i precedes v j ;

  2. Each arc including v 1 , v 2 satisfies that v 1 is the second last entry, v 2 is the last entry and the remaining k − 2 entries satisfy i < j if and only if v i precedes v j ;

  3. Each arc including v 1 but excluding v 2 satisfies that v 1 is the last entry and the remaining k − 1 entries satisfy i < j if and only if v i precedes v j ;

  4. Each arc including v 2 but excluding v 1 , v 3 satisfies that v 2 is the last entry and the remaining k − 1 entries satisfy i < j if and only if v i precedes v j ;

  5. Each arc including v 2 , v 3 but excluding v 1 satisfies that v 2 is the second last entry, v 3 is the last entry and the remaining k − 2 entries satisfy i < j if and only if v i precedes v j .

It is easy to check that T 3 is strong. Now we show that C 1 , 2 ( T 3 ) = K n − E ( P 3 ) . Note that N + ( v 1 ) = { v 2 } and N + ( v 2 ) = { v 3 } . By Corollary 2.2 ( a ) , we have v 1 v 2 , v 2 v 3 ∉ E ( C 1 , 2 ( T 3 ) ) . Now we consider the arc v i v j for { i , j } ≠ { 1 , 2 } and { i , j } ≠ { 2 , 3 } . W.l.o.g., we assume i < j .

  • For 3 ≤ i < j ≤ n − ( k − 3 ) , v i dominates v 2 by the arc ( v i , … , v i + ( k − 3 ) , v 1 , v 2 ) and v j dominates v 2 by the arc ( v j , … , v j + ( k − 3 ) , v 1 , v 2 ) . Then v i and v j compete and v i v j ∈ E ( C 1 , 2 ( T 3 ) ) .

  • For n − ( k − 4 ) ≤ i < j ≤ n , v i dominates v 2 by the arc ( v n − ( k − 2 ) , … , v i , … , v j , … , v n − 1 , v 1 , v 2 ) and v j dominates v 2 by the arc ( v n − ( k − 3 ) , … , v i , … , v j , … , v n , v 1 , v 2 ) . Then v i and v j compete and v i v j ∈ E ( C 1 , 2 ( T 3 ) ) .

  • For 3 ≤ i ≤ n − ( k − 3 ) and n − ( k − 4 ) ≤ j ≤ n , v i dominates v 2 by the arc ( v i , … , v i + ( k − 3 ) , v 1 , v 2 ) for 3 ≤ i ≤ n − ( k − 2 ) and by the arc ( v n − ( k − 2 ) , v n − ( k − 3 ) , … , v n − 1 , v 1 , v 2 ) for i = n − ( k − 3 ) and v j dominates v 2 by the arc ( v n − ( k − 3 ) , … , v j , … , v n , v 1 , v 2 ) . Then v i and v j compete and v i v j ∈ E ( C 1 , 2 ( T 3 ) ) .

  • For i = 1 and 3 ≤ j ≤ n − ( k − 2 ) , v 1 dominates v 2 by the arc ( v n − ( k − 3 ) , … , v n , v 1 , v 2 ) and v j dominates v 2 by the arc ( v j , … , v j + ( k − 3 ) , v 1 , v 2 ) . Then v 1 and v j compete and v 1 v j ∈ E ( C 1 , 2 ( T 3 ) ) .

  • For i = 1 and n − ( k − 3 ) ≤ j ≤ n , v 1 dominates v 2 by the arc ( v 3 , … , v k , v 1 , v 2 ) and v j dominates v 2 by the arc ( v n − ( k − 3 ) , … , v j , … , v n , v 1 , v 2 ) . Then v 1 and v j compete and v 1 v j ∈ E ( C 1 , 2 ( T 3 ) ) .

  • For i = 2 and 4 ≤ j ≤ n − ( k − 3 ) , v 2 dominates v 3 by the arc ( v n − ( k − 3 ) , … , v n , v 2 , v 3 ) , v 3 dominates v 1 by the arc ( v 3 , … , v k , v 1 , v 2 ) and v j dominates v 1 by the arc ( v j , … , v j + ( k − 3 ) , v 1 , v 2 ) . Then v 2 and v j ( 1 , 2 )-step compete and v 2 v j ∈ E ( C 1 , 2 ( T 3 ) ) .

  • For i = 2 and n − ( k − 4 ) ≤ j ≤ n , v 2 dominates v 3 by the arc ( v n − ( k − 3 ) , … , v n , v 2 , v 3 ) , v 3 dominates v 1 by the arc ( v 3 , … , v k , v 1 , v 2 ) and v j dominates v 1 by the arc ( v n − ( k − 3 ) , … , v j , … , v n , v 1 , v 2 ) . Then v 2 and v j ( 1 , 2 )-step compete and v 2 v j ∈ E ( C 1 , 2 ( T 3 ) ) .

Thus, C 1 , 2 ( T 3 ) = K n − E ( P 3 ) .

Now we show the “only if” part. Let T be a strong k-hypertournament and G the ( 1 , 2 ) -step competition graph of T. We show that G is K n , K n − E ( P 2 ) , or K n − E ( P 3 ) . We claim that G c contains at most two edges. Suppose to the contrary that G c contains at least three edges, say e 1 , e 2 , e 3 ∈ E ( G c ) . Let e i = x i y i for i = 1 , 2 , 3 . By Lemma 3.1, e 1 and e 2 have a common end-point. W.l.o.g., we assume that y 1 = x 2 . By Lemma 3.1, e 3 and e 1 have a common end-point, and e 3 and e 2 have also a common end-point. So either e 3 = x 1 y 2 or x 2 is an end-point of e 3 . However, this implies G c contains 3-cycle or K 1 , 3 , which contradicts Lemma 3.2 and Corollary 3.4. So G c contains at most two edges. Thus, if G c contains two edges, Lemma 3.1 implies G = K n − E ( P 3 ) ; if G c contains one edge, then G = K n − E ( P 2 ) ; if G c contains no edge, then G = K n .

Therefore, the theorem holds.□

5 Remaining k-hypertournaments

Theorem 5.1

A graph G on n vertices is the ( 1 , 2 ) -step competition graph of some k-hypertournament T with 3 ≤ k ≤ n − 1 if and only if G is K n , K n − E ( P 2 ) , K n − E ( P 3 ) , or K n − 1 ∪ K 1 .

Proof

The “if” part follows from the proof of Lemma 3.3 and Theorem 4.1. Now we show the “only if” part. Let T be a k-hypertournament and G the ( 1 , 2 ) -step competition graph of T. We show that G is K n , K n − E ( P 2 ) , K n − E ( P 3 ) , or K n − 1 ∪ K 1 . Similarly to the proof of “only if” of Theorem 4.1, we get G c contains at most two edges unless G = K n − 1 ∪ K 1 . Thus, if G c contains two edges, Lemma 3.1 implies G = K n − E ( P 3 ) ; if G c contains one edge, then G = K n − E ( P 2 ) ; if G c contains no edge, then G = K n .

Therefore, the theorem holds.□

6 The ( i , j ) -step competition graph of a k-hypertournament

We generalize the ( 1 , 2 ) -step competition graph to the ( i , j ) -step competition graph as follows. By the definition of the ( i , j ) -step competition graph for a k-hypertournament T, we obtain that if i ≥ 1 , j ≥ 2 , then E ( C 1 , 2 ( T ) ) ⊆ E ( C i , j ( T ) ) . It is easy to see that the proof of Lemma 2.1 implies the following corollary.

Corollary 6.1

Let T be a k-hypertournament with n vertices satisfying 3 ≤ k ≤ n − 1 and i ≥ 1 , j ≥ 2 integers. Then x y ∉ E ( C i , j ( T ) ) if and only if one of the following holds:

  1. N + ( x ) = ∅ ;

  2. N + ( y ) = ∅ ;

  3. N + ( x ) = { y } ;

  4. N + ( y ) = { x } ;

  5. A T ∗ { x , y } contains exactly an arc a , and N T − a + ( x ) ⊆ { y } , N T − a + ( y ) ⊆ { x } .

Theorem 6.2

Let T be a k-hypertournament with n vertices satisfying 3 ≤ k ≤ n − 1 and i ≥ 1 , j ≥ 2 integers. Then C i , j ( T ) = C 1 , 2 ( T ) .

Proof

Clearly, V ( C i , j ( T ) ) = V ( C 1 , 2 ( T ) ) = V ( T ) . Since E ( C 1 , 2 ( T ) ) ⊆ E ( C i , j ( T ) ) , it suffices to show that E ( C i , j ( T ) ) ⊆ E ( C 1 , 2 ( T ) ) . Let x y ∈ E ( C i , j ( T ) ) . Suppose x y ∉ E ( C 1 , 2 ( T ) ) . By Lemma 2.1, x and y must satisfy one of the cases (a)–(e). This contradicts Corollary 6.1. Thus, x y ∈ E ( C 1 , 2 ( T ) ) and E ( C i , j ( T ) ) ⊆ E ( C 1 , 2 ( T ) ) .□

  1. Funding information: This research was partially supported by the Youth Foundation of Shanxi Province (201901D211197).

  2. Conflict of interest: Authors state no conflict of interest.

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Received: 2020-10-12
Accepted: 2021-05-07
Published Online: 2021-06-10

© 2021 Ruijuan Li et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2021-0047
Keywords for this article
k-hypertournament; (1, 2)-step competition graph; (i, j)-step competition graph
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Articles in the same Issue

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  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
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  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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