Bulletin of the
Korean Mathematical Society BKMS

For a positive integer $m$ and a digraph $D$, the $m$-step competition graph $C^m(D)$ of $D$ has the same set of vertices as $D$ and an edge between vertices $u$ and $v$ if and only if there is a vertex $x$ in $D$ such that there are directed walks of length $m$ from $u$ to $x$ and from $v$ to $x$. Cho and Kim [6] introduced notions of competition index and competition period of $D$ for a strongly connected digraph $D$. In this paper, we extend these notions to a general digraph $D$. In addition, we study competition indices of tournaments.

Keywords: competition graph, $m$-step competition graph, competition index, competition period, tournament

MSC numbers: 05C20, 05C50