Adding a Tail in Classes of Perfect Graphs
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Consider a graph G which belongs to a graph class C. We are interested in connecting a node w ∉V(G) to G by a single edge u w where u ∈ 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 a minimum C-completion of G+w, i.e., the minimum number of non-edges (excluding the tail u w) 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.
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