Graphoidally Independent Infinite Cactus

A graphoidal cover of a graph $G$ (not necessarily finite) is a collection $\psi$ of paths (not necessarily finite, not necessarily open) satisfying the following axioms: (GC-1) Every vertex of $G$ is an internal vertex of at most one path in $\psi$, and (GC-2) every edge of $G$ is in exactly one path in $\psi$. The pair $(G, \psi)$ is called a graphoidally covered graph and the paths in $\psi$ are called the $\psi$-edges of $G$. In a graphoidally covered graph $(G, \psi)$, two distinct vertices $u$ and $v$ are $\psi$-adjacent if they are the ends of an open $\psi$-edge. A graphoidally covered graph $(G, \psi)$ in which no two distinct vertices are $\psi$-adjacent is called $\psi$-independent and the graphoidal cover $\psi$ is called a totally disconnecting graphoidal cover of $G$. Further, a graph possessing a totally disconnecting graphoidal cover is called a graphoidally independent graph. The aim of this paper is to establish complete characterization of graphoidally independent infinite cactus.