Safe separators for treewidth

A set of vertices S ⊆ 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 ∪ S. We show tha t 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 t o be safe, allowing such separators, if existing, to be found in polynomial t ime. In particular, every minimal separator of size one or two is safe, every minimal separator of size three tha t does not split off a component with only one vertex is safe, and every minimal separator tha t is an almost clique is safe; an almost clique is a set of vertices W such tha t there is a v ∈ W with W {v} a clique. We report on experiments tha t show significant reductions of instance sizes for graphs from probabilistic networks and frequency assignment.