The minimum completion (fill-in) problem is defined as follows: Given a graph family $\mathcal{F}$ (more generally, a property $\Pi$) and a graph $G$, the completion problem asks for the minimum number of non-edges needed to be added to $G$ so that the resulting graph belongs to the graph family $\mathcal{F}$ (or has property $\Pi$). This problem is NP-complete for many subclasses of perfect graphs and polynomial solutions are available only for minimal completion sets. We study the minimum completion problem of a $P_4$-sparse graph $G$ with an added edge. For any optimal solution of the problem, we prove that there is an optimal solution whose form is of one of a small number of possibilities. This along with the solution of the problem when the added edge connects two non-adjacent vertices of a spider or connects two vertices in different connected components of the graph enables us to present a polynomial-time algorithm for the problem.

翻译：最小完成（填充）问题定义如下：给定一个图族 $\mathcal{F}$（更一般地，一个性质 $\Pi$）和一个图 $G$，该完成问题要求添加最少的非边使 $G$ 转化为属于图族 $\mathcal{F}$（或具有性质 $\Pi$）的图。对于完美图的许多子类，该问题是NP完全的，且仅对最小完成集存在多项式解法。我们研究在$P_4$-稀疏图$G$中添加一条边的最小完成问题。针对该问题的任意最优解，我们证明存在一种最优解，其形式仅属于少量可能性之一。结合添加的边连接蜘蛛图中两个不相邻顶点或连接图中不同连通分量中的两个顶点时的解，我们为该问题提出了一种多项式时间算法。