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 ti⁻(G) 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≥3 and n≠5, then ti⁻(G)≤2, with equality if and only if n≡2(mod3). If G is a cubic graph of order n, then we show that ti⁻(G)≤[Formula presented]n. For r≥4 we show that if G is an r-regular graph of order n, then ti⁻(G)≤[Formula presented]n.