Safe Edges: A Study of Triangulation in Fill-in and Tree-Width Problems
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This paper considers two well-studied problems Minimum Fill-In (Min Fill-In) and Treewidth. Since both problems are -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 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 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 Min Fill-In and 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.
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