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

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.

Figures (1)

• Figure 1: The adjacency matrix $A$ of $C_{1,2}(D)$ for a multipartite tournament $D$ with a non-$\{1,2\}$-competing partite set $X_1$ where $U$ is the set of sinks in $D$ ($U$ is possibly vacuous); $F_i$, $O$, $J$, and $I$ stand for $\{v: \emptyset \neq N^+(v) \subseteq X_i\}$, a zero matrix, a matrix of all $1$'s, and an identity matrix, respectively; $X^*_1=X_1- \left(\bigcup_{i=2}^kF_i \cup U \right)$; $M$ is undetermined, yet, if $D$ has a sink or an anti-$\{1,2\}$-competing set of size at least three in a partite set, then $M=J-I$; Blocks marked with $\star$ depend on $D$.

Theorems & Definitions (34)

• Proposition 2.1: choi202412step
• Proposition 2.2: choi202412step
• Corollary 2.3: choi202412step
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4