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 m -step competition graph. In this paper, we offer another generalization of the competition graph. We define the ( 1 , 2 ) -step competition graph of a digraph D , denoted C 1 , 2 ( D ) , as the graph on V ( D ) where { x , y } ∈ E ( C 1 , 2 ( D ) ) if and only if there exists a vertex z ≠ x , y , such that either d D − y ( x , z ) = 1 and d D − x ( y , z ) ≤ 2 or d D − x ( y , z ) = 1 and d D − y ( x , z ) ≤ 2 . In this paper, we characterize the ( 1 , 2 ) -step competition graphs of tournaments and extend our results to the ( i , k ) -step competition graph of a tournament.