On Convex and Nonsplit Convex Domination in Graphs

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.