Consider a graph $G$ which belongs to a graph class ${\cal C}$. We are interested in connecting a node $w \not\in V(G)$ to $G$ by a single edge $u w$ where $u \in V(G)$; we call such an edge a \emph{tail}. As the graph resulting from $G$ after the addition of the tail, denoted $G+uw$, need not belong to the class ${\cal C}$, we want to compute a minimum ${\cal C}$-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 ${\cal C}$. 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.

翻译：考虑一个属于图类${\cal C}$的图$G$。我们关注如何通过单条边$uw$（其中$u \in V(G)$）将节点$w \not\in V(G)$连接到$G$，这样的边称为\emph{尾}。由于添加尾后得到的图$G+uw$可能不再属于图类${\cal C}$，我们希望计算$G+w$的最小${\cal C}$-完备化，即向$G+uw$中添加最少非边（不包括尾$uw$）使得结果图属于${\cal C}$。本文针对分割图、拟阈值图、阈值图和$P_4$-稀疏图类研究该问题，通过利用分割图的结构特性以及拟阈值图、阈值图和$P_4$-稀疏图的树表示，提出了线性时间算法。