68R10 Graph theory (including graph drawing) [See also 05Cxx, 90B10, 90B35, 90C35]
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A set of vertices $S\subseteq V$ is called a safe separator for treewidth, if $S$ is a separator of $G$, and the treewidth of $G$ equals the maximum of the treewidth over all connected components $W$ of $G-S$ of the graph, obtained by making $S$ a clique in the subgraph of $G$, induced by $W\cup S$. We show that such safe separators are a very powerful tool for preprocessing graphs when we want to compute their treewidth. We give several sufficient conditions for separators to be safe, allowing such separators, if existing, to be found in polynomial time. In particular, every minimal separator of size one or two is safe, every minimal separator of size three that does not split off a component with only one vertex is safe, and every minimal separator that is an almost clique is safe; an almost clique is a set of vertices $W$ such that there is a $v\in W$ with $W-\{v\}$ a clique. We report on experiments that show significant reductions of instance sizes for graphs from proba! bilistic networks and frequency assignment.
Several sets of reductions rules are known for preprocessing a graph when computing its treewidth. In this paper, we give reduction rules for a weighted variant of treewidth, motivated by the analysis of algorithms for probabilistic networks. We present two general reduction rules that are safe for weighted treewidth. They generalise many of the existing reduction rules for treewidth. Experimental results show that these reduction rules can significantly reduce the problem size for several instances of real-life probabilistic networks.