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, $\varnothing$, $V(\sigma )$ and $\{v\}$, for $v\in V(\sigma )$, are modules of $\sigma$ called trivial modules of $\sigma$. A 2-structure $\sigma$ is prime if $v(\sigma )\ge 3$ 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 $X⊊V(\sigma )$ 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.