Paper Info
Title
The structure of a 3-connected matroid with a 3-separating set of essential elements
Abstract
An element e of a 3-connected matroid M is essential if neither the deletion nor the contraction of e from M is 3-connected. Tutte's 1966 Wheels and Whirls Theorem proves that the only 3-connected matroids in which every element is essential are the wheels and whirls. It was proved by Oxley and Wu that if a 3-connected matroid M has a non-essential element, then it has at least two such elements. Moreover, the set of essential elements of M can be partitioned into classes where two elements are in the same class if M has a fan, a maximal partial wheel, containing both. In addition, if M has a fan with 2k or 2k + 1 elements for some k ≥ 2, then M can be obtained by sticking together a (k + 1)-spoked wheel and a certain 3-connected minor of M. In this paper, it is shown how a slight modification of these ideas can be used to describe the structure of a 3-connected matroid M having a 3-separation (A,B) such that every element of A is essential. The motivation for this study derives from a desire to determine when one can remove an element from M so as to both maintain 3-connectedness and preserve one side of the 3-separation.
Year
DOI
Venue
2003
10.1016/S0012-365X(02)00579-4
Discrete Mathematics
Keywords
Field
DocType
3-separating set,slight modification,essential element,3-connected matroids,whirls theorem,non-essential element,spoked wheel,element e,maximal partial wheel,3-connected matroid,05b35,theorem proving
Matroid,Discrete mathematics,Combinatorics,Matroid partitioning,Graphic matroid,Vertex separator,Mathematics
Journal
Volume
Issue
ISSN
265
1-3
Discrete Mathematics
Citations 
PageRank 
References 
2
0.67
6
Authors
1
Name
Order
Citations
PageRank
James Oxley119424.39