An inequality that relates the size of a bipartite graph with its order and restrained domination number

Let G=(V,E) be a graph. A set S⊆V is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and to a vertex in V-S. The restrained domination number of G, denoted γr(G), is the smallest cardinality of a restrained dominating set of G. Consider a bipartite graph G of order n≥4, and let k∈{2,3,..,n-2}. In this paper we will show that if γr(G)=k, then m≤((n-k)(n-k+6)+4k-8)/4. We will also show that this bound is best possible.