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

Abstract

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 C1,2(D){C}_{1,2}\left(D), is a graph on V(D)V\left(D), where xy∈E(C1,2(D))xy\in E\left({C}_{1,2}\left(D)) if and only if there exists a vertex z≠x,yz\ne x,y such that either dD−y(x,z)=1{d}_{D-y}\left(x,z)=1 and dD−x(y,z)≤2{d}_{D-x}(y,z)\le 2 or dD−x(y,z)=1{d}_{D-x}(y,z)=1 and dD−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.

Keywords

• k-hypertournament
• (1, 2)-step competition graph
• (i, j)-step competition graph
• 05c65
• 05c12
• 05c20