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

TL;DR

This paper investigates the structure of $(1,2)$-step competition graphs of multipartite tournaments, extending previous work to include non-tight cases, and explores properties like interval and connectivity.

Contribution

It characterizes $(1,2)$-step competition graphs for loose multipartite tournaments, broadening understanding beyond tight cases.

Findings

Characterization of $C_{1,2}(D)$ for loose multipartite tournaments.

Conditions under which $C_{1,2}(D)$ is interval.

Conditions for $C_{1,2}(D)$ to be connected.

Abstract

A multipartite tournament is an orientation of a complete $k$-partite graph for some positive integer $k\geq 3$. 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 the previous paper titled "On $(1,2)$-step competition graphs of multipartite tournaments" \cite{choi202412step} we completely characterize $C_{1,2}(D)$ for a tight multipartite tournament $D$. As an extension, in this paper, we study $(1,2)$-step competition graphs of multipartite tournaments that are not tight, which will be called loose. For a loose multipartite tournament $D$, various meaningful results are obtained in terms of $C_{1,2}(D)$ being interval and $C_{1,2}(D)$ being connected.