The induced path number \( \rho(G) \) 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. A Nordhaus-Gaddum type result is a (tight) lower or upper bound on the sum (or product) of a parameter of a graph and its complement. If \( G \) is a subgraph of \( H \), then the graph \( H – E(G) \) is the complement of \( G \) relative to \( H \). In this paper, we consider Nordhaus-Gaddum type results for the parameter \( \rho \) when the relative complement is taken with respect to the complete bipartite graph \( K_{m,n} \).