Total-rainbow connection and forbidden subgraphs

A path in a total-colored graph $G$ is called a total-rainbow path if its edges and internal vertices have distinct colors. The total-colored graph $G$ is total-rainbow connected if for any two distinct vertices of $G$, there is a total-rainbow path connecting them. The total-rainbow connection number of $G$, denoted as $trc\left(G\right)$, represents the minimum number of colors that are required to make $G$ total-rainbow connected. This paper characterizes all the families $F$ of connected graphs with $\left|F\right|\in \{1,2\}$, for which there exists a constant $k_F$ such that $G$ being a connected $F$-free graph implies $trc\left(G\right)\le 2diam\left(G\right)+k_F$, where $diam\left(G\right)$ denotes the diameter of $G$.