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

中文翻译：

---

超级锦标赛的 (1, 2) 步竞争图

2011 年，Factor 和 Merz [Discrete Appl. 数学。159 (2011), 100–103] 定义了有向图的 ( 1 , 2 ) \left(1,2) -step 竞争图。给定一个有向图 D = ( V , A ) D=\left(V,A) ， D 的 ( 1 , 2 ) \left(1,2) 步竞争图，表示为 C 1 , 2 ( D ) {C }_{1,2}\left(D) , 是 V ( D ) V\left(D) 上的图，其中 xy ∈ E ( C 1 , 2 ( D ) ) xy\in E\left({C }_{1,2}\left(D)) 当且仅当存在顶点 z ≠ x , yz\ne x,y 使得 d D − y ( x , z ) = 1 {d}_{ Dy}\left(x,z)=1 且 d D − x ( y , z ) ≤ 2 {d}_{Dx}(y,z)\le 2 或 d D − x ( y , z ) = 1 {d}_{Dx}(y,z)=1 和 d D − y ( x , z ) ≤ 2 {d}_{Dy}\left(x,z)\le 2 。他们还对锦标赛的 (1, 2) 步竞争图进行了表征，并将一些结果扩展到锦标赛的 ( i , j ) \left(i,j) 步竞争图。在本文中，(1, 2)-step 有向图的竞争图被推广到一个超级锦标赛，并且表征了 ak -超级锦标赛的 (1, 2)-step 竞争图。此外，结果被扩展到 ( i , j ) \left(i,j) -k 超级锦标赛的步竞争图。