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In this paper, the concepts of extension of a graph(digraph) and the extensible class of graphs(digraphs) have been introduced. The class of connected graphs as well as the class of Hamiltonian graphs which are extensible classes have also been proved. The classes of regular, eulerian, bipartite and trees graphs which are not extensible classes have also been proved.  The concept of extensibility number has been introduced as well as the characterization of regular graphs(digraphs) which have extensibility number  k . Also the extensibility number of eulerian graphs(digraphs) has been characterized.