Partially critical 2-structures

Abstract

A 2-structure $\sigma$ consists of a vertex set $V(\sigma)$ and of an equivalence relation $\equiv_\sigma$ defined on $(V(\sigma)\times V(\sigma))\setminus\{(v,v):v\in V(\sigma)\}$. Given a 2-structure $\sigma$, a subset $M$ of $V(\sigma)$ is a module of $\sigma$ if for $x,y\in M$ and $v\in V(\sigma)\setminus M$, $(x,v)\equiv_{\sigma}(y,v)$ and $(v,x)\equiv_{\sigma}(v,y)$. For instance, $\emptyset$, $V(\sigma)$ and $\{v\}$, for $v\in V(\sigma)$, are modules of $\sigma$ called trivial modules of $\sigma$. A 2-structure $\sigma$ is prime if $v(\sigma)\geq 3$ and all the modules of $\sigma$ are trivial. A prime 2-structure $\sigma$ is critical if for each $v\in V(\sigma)$, $\sigma-v$ is not prime. A prime 2-structure $\sigma$ is partially critical if there exists $X\subsetneq V(\sigma)$ such that $\sigma[X]$ is prime, and for each $v\in V(\sigma)\setminus X$, $\sigma-v$ is not prime. We characterize finite or infinite partially critical 2-structures.