Constructing New Disconnected Domination Parameter in Graphs

Résumé

In this paper, a disconnected domination model in graph theory, known as doubly disconnected domination, is introduced. Suppose that  be a simple, undirected, finite and nontrivial graph. A subset  is called doubly disconnected dominating set in  if  is dominating set and both the induced subgraphs  and  are disconnected subgraphs. The least cardinality among all doubly disconnected dominating sets of  is the doubly disconnected domination number . This study investigates various bounds and properties of this domination parameter to give the relations between  and the maximum degree in , minimum degree, the size and the order of the graph. Some results are given in this paper to explain and prove the doubly disconnected domination number  for any graph constructed by corona or join operations. Furthermore,  is discussed and evaluated for some well-known graphs, while other graphs are proved to be hasn't this type of domination.