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

The competition graph of a digraph, introduced by Cohen in 1968, has been extensively studied. More recently, in 2000, Cho, Kim, and Nam defined the mm-step competition graph. In this paper, we offer another generalization of the competition graph. We define the (1,2)(1,2)-step competition graph of a digraph DD, denoted C1,2(D)C1,2(D), as the graph on V(D)V(D) where {x,y}∈E(C1,2(D)){x,y}∈E(C1,2(D)) if and only if there exists a vertex z≠x,yz≠x,y, such that either dD−y(x,z)=1dD−y(x,z)=1 and dD−x(y,z)≤2dD−x(y,z)≤2 or dD−x(y,z)=1dD−x(y,z)=1 and dD−y(x,z)≤2dD−y(x,z)≤2. In this paper, we characterize the (1,2)(1,2)-step competition graphs of tournaments and extend our results to the (i,k)(i,k)-step competition graph of a tournament.