Adding a Tail in Classes of Perfect Graphs

Consider a graph G which belongs to a graph class C. We are interested in connecting a node w is not an element of V (G) to G by a single edge uw where u is an element of V (G); we call such an edge a tail. As the graph resulting from G after the addition of the tail, denoted G + uw, need not belong to the class C, we want to compute the number of non-edges of G in a minimum C -completion of G + uw, i.e., the minimum number of non-edges (excluding the tail uw) to be added to G + uw so that the resulting graph belongs to C. In this paper, we study this problem for the classes of split, quasi-threshold, threshold and P-4-sparse graphs and we present linear-time algorithms by exploiting the structure of split graphs and the tree representation of quasi-threshold, threshold and P-4-sparse graphs.