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

Abstract

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.