On defect sets in bipartite graphs (extended abstract)

Abstract

A defect set in a bipartite graph with vertex classes V and W is a subset X ⊂ V such that the neighbourhood N(X) satisfies |N(X)| < |X|. We study a lemma on defect sets in bipartite graphs with certain expanding properties from the algorithmic complexity point of view. This lemma is the core of a result of Friedman and Pippenger which states that expanding graphs contain all small trees. We also discuss related problems of finding shortest circuits of matroids represented over a field. In particular, we propose a new straightforward method to derive a weaker form (PR-completeness) of the recent NP-completeness results of Khachiyan [11] and Vardy [18] concerning this problem for the field of rationals and GF(p m), respectively.KeywordsPolynomial TimeDecision ProblemBipartite GraphShort CircuitFinite FieldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.