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

Competition graphs were created in connected to a biological model as a means of reflecting the competition relations among the predators in the food webs and determining the smallest dimension of ecological phase space. In 2011, Factor and Merz introduced 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 $xy\in E(C_{1,2}(D))$ if and only if there exists a vertex $z\neq x,y$ such that either $d_{D-y}(x,z)=1$ and $d_{D-x}(y,z)\leq 2$ or $d_{D-x}(y,z)=1$ and $d_{D-y}(x,z)\leq 2$. 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 the one of a hypertournament and the $(1,2)$-step competition graph of a $k$-hypertournament is characterized. Also, the results are extended to the $(i,j)$-step competition graph of a $k$-hypertournament.

. 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.
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][3][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. 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: 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 ≤ ≤ − k n 3 1 .
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 ā denotes the set of vertices contained in a. If ∈ v v a ,ī j 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: 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 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.
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: , 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.
is also called the competition graph of the k-hypertournament T. Clearly, when = k 2, T is a tournament and ( ) C T i j , 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][8][9][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, 2step 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 if and only if one of the following holds: contains exactly an arc a, and Proof. First, we show the "if" part. Clearly, if one of (a)-(d) holds, we have ∉ ( ( )) xy E C T 1,2 . Now we assume that the argument (e) holds. Since { } * A x y , T contains exactly an arc a, and , 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 . Also, assume that x and y do not satisfy if and only if one of the following holds: contains exactly an arc a, and 3 The forbidden subgraphs of ( ( )) C T c 1,2 Suppose that at least one of xy and zw 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 ∈ ( ( )) = ( ) zw E C T E G , 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 z w , T contains at least two arcs, which contradicts the fact that zw satisfies ( ) e . The lemma holds. □   Proof. Suppose at least two edges among xy, xz and yz satisfy the case ( ) e . W.l.o.g., we assume that both xz and yz satisfy ( ) e . From the assumption that xz satisfies ( ) e , we get the vertex y dominates x in each arc containing x y , but not containing z. From the assumption that yz 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 xy, xz and yz satisfies the case ( ) e . □

∉ ( ) xy xz yz E G
, , hold simultaneously. Thus, the complement G c of G does not contain 3-cycle. The lemma holds. □ Proof. Suppose at least one of xy, xz and xw satisfies the case ( ) a or ( ) b . W.l.o.g., we assume that xy satisfies ( ) a , i.e. ( ) = ∅ , , .
Then v i and v j compete and .
Then v i and v j compete and Proof. Suppose at least two edges among xy, xz and xw satisfy the case ( ) c or ( ) d . W.l.o.g., we assume that both xy and xz satisfy the case ( ) c or ( ) d . We consider the following four cases. Proof. Suppose at least two edges among xy, xz and xw satisfy the case ( ) e . Assume that xy and xz satisfy ( ) e . From the assumption that xy satisfies ( ) e , we get the vertex z dominates y in each arc containing y z , but not containing x. From the assumption that xz satisfies ( ) e , we get the vertex y dominates z in each arc containing y z , but not containing x. This is a contradiction.  Proof. We first show the "if" part. Let T be a transitive k-hypertournament with the vertices Let T 1 be a k-hypertournament obtained from T by replacing the arc ( … ) , , , i n k n n 2 1 and v j dominates v n by the arc , , , Let T 2 be a k-hypertournament obtained from T 1 above by replacing the arc ( … ) , , (2) Each arc including v v , 1 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 v , 1 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 v , 2 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 T K E P . It is easy to see that the proof of Lemma 2.1 implies the following corollary.

Conflict of interest:
Authors state no conflict of interest.