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 $sqrt n leqslant rho left( G right) + rho left( {bar G} right) leqslant leftlceil {tfrac{{3n}} {2}} rightrceil$ . In this paper, we characterize the graphs G for which $rho left( G right) + rho left( {bar G} right) = leftlceil {tfrac{{3n}} {2}} rightrceil$ , 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.