Reducing regular graphs to partition their vertices into a total dominating set and an independent dominating set

A graph $G$ whose vertex set can be partitioned into a total dominating set and an independent dominating set is called a TI-graph. There exist infinite families of graphs that are not TI-graphs. We define the TI-reduction number ${\mathrm{ti}}^{-}\left(G\right)$ of a graph $G$ to be the minimum number of edges that must be removed from $G$ to ensure that the resulting graph is a TI-graph. We observe that the TI-reduction number exists for all connected graphs with minimum degree at least 2 with one exception, namely the 5-cycle. We show that if $G$ is a cycle $C_n$ with $n\ge 3$ and $n\ne 5$, then ${\mathrm{ti}}^{-}\left(G\right)\le 2$, with equality if and only if $n\equiv 2\left(\mathrm{mod}3\right)$. If $G$ is a cubic graph of order $n$, then we show that ${\mathrm{ti}}^{-}\left(G\right)\le \frac{1}{4}n$. For $r\ge 4$ we show that if $G$ is an $r$-regular graph of order $n$, then ${\mathrm{ti}}^{-}\left(G\right)\le \frac{1}{2}(r-1)\left(\frac{r}{3r-1}\right)n$.