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

Abstract

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 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 $\lceil \frac{n}{4} \rceil \le \rho (G) \rho (\overline{G}) \le n \lceil \frac{n}{2} \rceil$, where these bounds are best possible. It was also noted that the upper bound is achieved when either G or $\overline{G}$ is a graph consisting of n isolated vertices. In this paper, we determine best possible upper and lower bounds for $\rho (G) \rho (\overline{G})$ when either both G and $\overline{G}$ are connected or neither G nor $\overline{G}$ has isolated vertices.