This paper considers two well-studied problems \textsc{Minimum Fill-In} (\textsc{Min Fill-In}) and \textsc{Treewidth}. Since both problems are \textsf{NP}-hard, various reduction rules simplifying an input graph have been intensively studied to better understand the structural properties relevant to these problems. Bodlaender at el. introduced the concept of a safe edge that is included in a solution of the \textsc{Minimum Fill-In} problem and showed some initial results. In this paper, we extend their result and prove a new condition for an edge set to be safe. This in turn helps us to construct a novel reduction tool for \textsc{Min Fill-In} that we use to answer other questions related to the problem. In this paper, we also study another interesting research question: Whether there exists a triangulation that answers both problems \textsc{Min Fill-In} and \textsc{Treewidth}. To formalise our study, we introduce a new parameter reflecting a distance of triangulations optimising both problems. We present some initial results regarding this parameter and study graph classes where both problems can be solved with one triangulation.

翻译：本文研究了两个经典问题：最小填充（\textsc{Minimum Fill-In}，简称\textsc{Min Fill-In}）与树宽（\textsc{Treewidth}）。由于这两个问题均为\textsf{NP}-难问题，为深入理解与这些问题相关的结构性质，研究者们已广泛探索各类简化输入图的归约规则。Bodlaender等人引入了安全边的概念，即包含在最小填充问题解中的边，并给出了若干初步结果。本文拓展了他们的结论，提出了边集安全性的新判定条件。这有助于我们构建一种针对最小填充问题的新型归约工具，并应用于该问题其他相关问题的解答。此外，本文还探讨了另一个有趣的研究问题：是否存在一种三角剖分能同时解决最小填充与树宽两个问题？为形式化研究，我们引入了一个新参数，用以反映同时优化这两个问题的三角剖分之间的距离。我们展示了该参数的初步结果，并研究了可通过单一三角剖分解得这两个问题的图类。