Morphologie des posets (-1)-critiques

Abstract

Let $G=(V,A)$ be a digraph. For $X\subseteq V$, the subdigraph of $G$ induced by $X$ is denoted by $G[X]$. A subset $I$ of $V$ is an interval of $G$ if for every $a,b \in I$ and $x \in V \setminus I$, $(x,a) \in A$ if and only if $(x,b) \in A$, and similarly for $(a,x)$ and $(b,x)$. The trivial intervals of $G$ are $\varnothing$, $V$ and $\lbrace x\rbrace$, where $x\in V$. The digraph $G$ is indecomposable if $| V(G)|\geqslant 3$ and all its intervals are trivial. Given an indecomposable digraph $G$, a vertex $x$ of $G$ is critical, if the induced subdigraph $G[V(G) \setminus \{x\}]$ is decomposable. The digraph $G$ is said to be (-1)-critical if it admits a single non-critical vertex. A poset (or a strict partial order) is a transitive digraph. In this paper, We characterize the (-1)-critical posets.