A Nordhaus-Gaddum-type result for the induced path number

The induced path number ??(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 graph. 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 ?? when the relative complement is taken with respect to the complete bipartite graph K _n,n .