  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.
