In the context of a finite undirected graph D( V,E) and a non-empty subset σ⊆ V, the graph generated by switching D by σ is denoted as D σ ( V , E ′ ) . This graph is acquired from D by summing up all non-edges that connect σ to the vertices V− σ and terminating all edges that connect σ and its complement V− σ. We record D v for σ={ v}, and the associated switching is known as vertex switching. Nevertheless, we refer to this as | σ|-vertex switching. It is reported as 2-vertex switching when | σ|=2. A two-cyclic graph contains exactly two cycles in it. Any two vertices in a connected graph are linked together by a path. In this article, we provide neccessary and sufficient requirements for D σ , the switching of D at σ={ e,f} to be connected and two cyclic graph when ef∈ E( D).