The Open Neighborhood Number of a Graph

Let G be a graph. A subset S of vertices in a graph G is an open neighborhood set if G =∪_(v∈S) 〈 N(v) 〉 where N(v) denotes an open neighborhood of a vertex v. The minimum cardinality of an open neighborhood set is called the open neighborhood number of a graph, denoted by n_op (G). In this paper, we initiate the study of the open neighborhood number. We determine this number for some standard family of graphs and some bounds are obtained. Further we study the effect of the operation maximum degree based vertex addition on this parameter.