Abstract
We show that the problems of finding a graph's vertex separation number and path-width are the same. The equivalence of these problems with the gate matrix layout and the node search number problems then follows immediately from the results of Fellows and Langston and Kirousis and Papadimitriou, respectively. The fixed parameter variants of problems are known to possess O( n 2) decision algorithms based on finite but unknown obstruction sets. We show how all tree obstructions in these sets may be constructively obtained.
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