Sharp Nordhaus-Gaddum bounds for throttling

Throttling is a graph optimization problem, where the throttling number of a graph is the minimum sum or minimum product of the number of vertices in an initial set and the time required to complete a certain graph operation. A Nordhaus-Gaddum bound refers to an upper or lower bound of the sum or product of a graph parameter together with that of its complement. In this paper, we study the Nordhaus-Gaddum sum and product bounds of the various throttling numbers (sum throttling and product throttling with or without initial cost). Graph operations considered are standard zero forcing, positive semidefinite forcing, power domination, and Cops and Robbers.