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

Abstract

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