Best possible upper bounds on the restrained domination number of cubic graphs

A dominating set in a graph (Formula presented.) is a set (Formula presented.) of vertices such that every vertex in (Formula presented.) is adjacent to a vertex in (Formula presented.). A restrained dominating set of (Formula presented.) is a dominating set (Formula presented.) with the additional restraint that the graph (Formula presented.) obtained by removing all vertices in (Formula presented.) is isolate-free. The domination number (Formula presented.) and the restrained domination number (Formula presented.) are the minimum cardinalities of a dominating set and restrained dominating set, respectively, of (Formula presented.). Let (Formula presented.) be a cubic graph of order (Formula presented.). A classical result of Reed states that (Formula presented.), and this bound is best possible. To determine the best possible upper bound on the restrained domination number of (Formula presented.) is more challenging, and we prove that (Formula presented.).