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.