A set \(D \subseteq V\) of a graph \(G =(V, E)\) is called a dominating set of G if every vertex in \(V\setminus D\) is adjacent to at least one vertex in D. A dominating set D of a graph G is convex dominating set if all vertices from \(u-v\) geodesic belong to D for every two vertices \(u,v \in D\) . A convex dominating set D of a graph G is nonsplit convex dominating set if the induced subgraph \(G[V \setminus D]\) is connected. The nonsplit convex domination number of G is the minimum cardinality of a nonsplit convex dominating set D and it is denoted by \(\gamma _{nscon}(G)\) . In this paper, we initiate the study on this parameter. We establish bounds for nonsplit convex domination number, \(\gamma _{nscon}(G)\) , of standard graph structures. Further, we also present conditions for identifying or constructing a nonsplit convex dominating set in any connected graph G.
