Nordhaus–Gaddum Results for the Induced Path Number of a Graph When Neither the Graph Nor Its Complement Contains Isolates

The induced path number $$\rho (G)$$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. A product Nordhaus–Gaddum-type result is a bound on the product of a parameter of a graph and its complement. Hattingh et al. (Util Math 94:275–285, 2014) showed that if G is a graph of order n, then $$\lceil \frac{n}{4} \rceil \le \rho (G) \rho (\overline{G}) \le n \lceil \frac{n}{2} \rceil $$⌈n4⌉≤ρ(G)ρ(G¯)≤n⌈n2⌉, where these bounds are best possible. It was also noted that the upper bound is achieved when either G or $$\overline{G}$$G¯ is a graph consisting of n isolated vertices. In this paper, we determine best possible upper and lower bounds for $$\rho (G) \rho (\overline{G})$$ρ(G)ρ(G¯) when either both G and $$\overline{G}$$G¯ are connected or neither G nor $$\overline{G}$$G¯ has isolated vertices.