In this paper the concept of reducibility in graph theory is discussed, and the deletable vertex (edge) in graph (digraph) is defined. The class of graphs (digraphs) ℛ  is called vertex (edge) reducible if for any G∈ ℛ  either ℛ is the trivial graph (null graph) or it contains a vertex (edge) v such that G-v∈ ℛ. We introduce some classes of graphs (digraphs) which are reducible and others which are not. The vertex reducibility and edge reducibility of Eulerian graphs and Eulerian digraphs have also been studied