Given a property (graph class) $\Pi$, a graph $G$, and an integer $k$, the \emph{$\Pi$-completion} problem consists in deciding whether we can turn $G$ into a graph with the property $\Pi$ by adding at most $k$ edges to $G$. The $\Pi$-completion problem is known to be NP-hard for general graphs when $\Pi$ is the property of being a proper interval graph (PIG). In this work, we study the PIG-completion problem %when $\Pi$ is the class of proper interval graphs (PIG) within different subclasses of chordal graphs. We show that the problem remains NP-complete even when restricted to split graphs. We then turn our attention to positive results and present polynomial time algorithms to solve the PIG-completion problem when the input is restricted to caterpillar and threshold graphs. We also present an efficient algorithm for the minimum co-bipartite-completion for quasi-threshold graphs, which provides a lower bound for the PIG-completion problem within this graph class.

翻译：给定一个性质（图类）$\Pi$、一个图 $G$ 和一个整数 $k$，\emph{$\Pi$-补全}问题旨在判断是否可以通过向 $G$ 添加至多 $k$ 条边，将其转化为具有性质 $\Pi$ 的图。当 $\Pi$ 为适当区间图（PIG）性质时，$\Pi$-补全问题在一般图上已知为 NP-困难。本文研究弦图的不同子类中的 PIG-补全问题。我们证明，即使限制在分裂图上，该问题仍保持 NP-完全性。随后，我们关注积极结果，并针对输入限制为毛毛虫图和阈值图的情况，提出多项式时间算法来求解 PIG-补全问题。此外，针对拟阈值图的最小共二部图补全问题，我们提出一种高效算法，该算法为该图类中的 PIG-补全问题提供了下界。