On The Trees Whose \(2\)-Step Compitition Numbers are Two

Abstract

Since Cohen introduced the notion of competition graph in \(1968\), various variations have been defined and studied by many authors. Using the combinatorial properties of the adjacency matrices of digraphs, Cho \(et\; al\). \([2]\) introduced the notion of a \(m\)-step competition graph as a generalization of the notion of a competition graph. Then they \([3]\) computed the \(2\)-step competition numbers of complete graphs, cycles, and paths. However, it seems difficult to compute the \(2\)-step competition numbers even for the trees whose competition numbers can easily be computed. Cho \(et\; al\). \([1]\) gave a sufficient condition for a tree to have the \(2\)-step competition number two. In this paper, we show that this sufficient condition is also a necessary condition for a tree to have the \(2\)-step competition number two, which completely characterizes the trees whose \(2\)-step competition numbers are two. In fact, this result turns out to characterize the connected triangle-free graphs whose \(2\)-step competition numbers are two.