Weak rainbow saturation numbers of paths, stars and cycles

An edge-colored graph is \emph{rainbow} if all of its edges receive distinct colors. For a fixed graph $H$, an edge-colored graph $F$ is called weakly $H$-rainbow saturated if there exists an ordering $e_1,e_2,\ldots,e_{|E(\bar{F})|}$ of $E(\bar{F})$ such that, for any edge coloring $c$ of $E(\bar{F