On the (co)girth of a connected matroid

  • Authors:

  • Jung Jin Cho;Yong Chen;Yu Ding

  • Affiliations:

  • Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX 77843, USA;Department of Mechanical and Industrial Engineering, The University of Iowa, Iowa City, IA 52242, USA;Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX 77843, USA

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2007

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Abstract

This article studies the girth and cogirth problems for a connected matroid. The problem of finding the cogirth of a graphic matroid has been intensively studied, but studies on the equivalent problem for a vector matroid or a general matroid have been rarely reported. Based on the duality and connectivity of a matroid, we prove properties associated with the girth and cogirth of a matroid whose contraction or restriction is disconnected. Then, we devise algorithms that find the cogirth of a matroid M from the matroids associated with the direct sum components of the restriction of M. As a result, the problem of finding the (co)girth of a matroid can be decomposed into a set of smaller sub-problems, which helps alleviate the computation. Finally, we implement and demonstrate the application of our algorithms to vector matroids.