The Non-Isolated Domination Number of a Graph

A subset S of the vertex set V (G) of a graph G is said to be a dominating set if every vertex not in S is adjacent to at least one vertex in S. In this research, we introduce a new domination parameter called the non-isolated domination number of a graph. A subset S of V of a nontrivial graph G is said to be a non-isolated dominating set if S is a dominating set and there are no zero-degree vertices in the subgraph induced by S. The minimum cardinality taken over all non-isolated dominating sets is called the non-isolated domination number and is denoted by γI. In this research, we obtained lower and upper bounds for the non-isolated domination number of a connected graph. We also determine the characterization of connected graphs that have the non-isolated domination numbers 2 and 3. Furthermore, we determine the non-isolated domination number of complete, n-partite complete, wheel, fan, star, cycle, and path graphs. We also determine the characterization of tree graphs that have the non-isolated domination number 2γ.