Infinite graphs with finite dominating sets

In this paper, we study the infinite graphs which admit a finite dominating set. The main contribution of the work is two folds: (i) characterization of infinite trees and hence infinite connected graphs with a finite dominating set, (ii) it is shown that apart from a family of graphs, all infinite graphs or their complements possess a finite dominating set. Moreover, some conditions of existence of finite dominating sets in product graphs are studied.