Nordhaus-Gaddum results for the sum of the induced path number of a graph and its complement

The induced path number ρ(G) of a graph G is defined as the minimum number of subsets into which the vertex set of G can be partitioned so that each subset induces a path. Broere et al. proved that if G is a graph of order n, then $\sqrt n \leqslant \rho \left( G \right) + \rho \left( {\bar G} \right) \leqslant \left\lceil {\tfrac{{3n}} {2}} \right\rceil$ . In this paper, we characterize the graphs G for which $\rho \left( G \right) + \rho \left( {\bar G} \right) = \left\lceil {\tfrac{{3n}} {2}} \right\rceil$ , improve the lower bound on $\rho \left( G \right) + \rho \left( {\bar G} \right)$ by one when n is the square of an odd integer, and determine a best possible upper bound for $\rho \left( G \right) + \rho \left( {\bar G} \right)$ when neither G nor $\bar G$ has isolated vertices.