The $(-1)$-Critically Duo-Free Tournaments

Abstract

Given a tournament $T=(V,A)$, a subset $X$ of $V$ is an interval of $T$ provided that for any $a,b\in X$ and $x\in V\setminus X$, $(a,x)\in A$ if and only if $(b,x)\in A$. For example, $\mathrm{\varnothing }$, $\{x\}$ ($x\in V$), and $V$ are intervals of $T$, called trivial intervals. A two-element interval of $T$ is called a duo of $T$. Tournaments that do not admit any duo are called duo-free tournaments. A vertex $x$ of a duo-free tournament is $d$-critical if $T–x$ has at least one duo. In 2005, J.F. Culus and B. Jouve [5] characterized the duo-free tournaments, all of whose vertices are d-critical, called tournaments without acyclic interval. In this paper, we characterize the duo-free tournaments that admit exactly one non-d-critical vertex, called (-1)-critically duo-free tournaments.