The m-step, same-step, and any-step competition graphs

The competition graph and its generalizations have interested graph theorists for over 30 years. Recently, Cho, Kim, and Nam introduced the m-step competition graph and computed the 2-step competition numbers of paths and cycles. We extend their results in a partial determination of the m-step competition numbers of paths and cycles. In addition, we introduce two new variants of competition graphs: same-step and any-step. We classify same-step and any-step competition graphs and investigate their related competition numbers.