The splitting number of the complete graph (Q1821790)

Let v be a vertex of a graph with adjacent vertices \(v_ 1,...,v_ k\), \(v_{k+1},...,v_ n\). Define a new graph \(G^*\) where v is replaced by \(v^*\) and \(v^{**}\). Except for \(v^*\) and \(v^{**}\), the vertices and adjacencies of \(G^*\) are as in G-v. The vertices adjacent to \(v^*\) are \(v_ 1,...,v_ k\), and those adjacent to \(v^{**}\) are \(v_{k+1},...,v_ n\). We say that \(G^*\) has been obtained from G by a vertex split. Define \(\sigma\) (G,S) to be the smallest number of vertex splits needed to make G to embeddable in the surface S. The main results of the paper are the determination of \(\sigma\) (G,S), where G is a complete graph and S is the plane, projective plane, or Klein bottle.