On $(1,2)$-step competition graphs of multipartite tournaments

Abstract

A multipartite tournament is an orientation of a complete $k$-partite graph for some positive integer $k\ge 3$. We say that a multipartite tournament $D$ is tight if every partite set forms a clique in the $(1,2)$-step competition graph, denoted by $C_{1,2}(D)$, of $D$. In this paper, we completely characterize $C_{1,2}(D)$ for a tight multipartite tournament $D$. We will study $C_{1,2}(D)$ for a multipartite tournament $D$ that is not tight in a follow up paper.