On defect sets in bipartite graphs

Abstract

A defect set in a bipartite graph with vertex classes V and W is a subset X C 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 temma 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 (Pit-completeness) of the recent NP-completeness results of Khachlyan [11] and Vardy [18] concerning this problem for the field of rationals and GF(pm), respectively.