On the (co)girth of a connected matroid

• Published in: Discrete Applied Mathematics Vol. 155; no. 18; pp. 2456 - 2470
• Main Authors: Cho, Jung Jin, Chen, Yong, Ding, Yu
• Format: Journal Article
• Language: English
• Published: Lausanne Elsevier B.V 01-11-2007; Amsterdam Elsevier; New York, NY
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Cogirth; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Convex and discrete geometry; Exact sciences and technology; Geometry; Girth; Mathematics; Matroid connectivity; Matroids; Sciences and techniques of general use; Theoretical computing; Cogirth; Matroids; Girth; Matroid connectivity; Computer theory; Matroid; Duality; Algorithm; Contraction; Implementation; Optimization; Graphics; Graph connectivity; Combinatorics; Application; Vector
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2007.06.015

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.