Self-switching of union of two complete graphs

By a graph H = (V, E), we mean a finite undirected graph without loops and multiple edges. Let H be a graph and σ ⊆ V be a non–empty subset of V. Hσ is the graph obtained from H by removing all edges between σ and its complement V-σ and adding as edges all non-edges between σ and V-σ. Then σ is said to be a self-switching of H if H ≅ Hσ. It can also be referred to as k-vertex self-switching where k = |σ|. The set of all self-switchings of the graph H with cardinality k is represented by SSk(H) and its cardinality by ssk(H). A graph on m vertices in which each pair of distinct vertices are neighbors is called a complete graph and is denoted by Km. Km∪ Kn is the union of two complete graphs and is disconnected. In this paper, we give necessary and sufficient conditions for σ to be a self-switching for the graph H=Km ∪ Kn and using this, we find the cardinality ssk(H).