The Structure of the 4-separations in 4-connected Matroids

نویسندگان

  • JEREMY AIKIN
  • JAMES OXLEY
چکیده

For a 2-connected matroid M , Cunningham and Edmonds gave a tree decomposition that displays all of its 2-separations. When M is 3-connected, two 3-separations are equivalent if one can be obtained from the other by passing through a sequence of 3-separations each of which is obtained from its predecessor by moving a single element from one side of the 3-separation to the other. Oxley, Semple, and Whittle gave a tree decomposition that displays, up to this equivalence, all nontrivial 3-separations of M . Now let M be 4-connected. In this paper, we define two 4-separations of M to be 2-equivalent if one can be obtained from the other by passing through a sequence of 4-separations each obtained from its predecessor by moving at most two elements from one side of the 4-separation to the other. The main result of the paper proves that M has a tree decomposition that displays, up to 2-equivalence, all non-trivial 4-separations of M .

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

The structure of 2-separations of infinite matroids

Generalizing a well known theorem for finite matroids, we prove that for every (infinite) connected matroid M there is a unique tree T such that the nodes of T correspond to minors of M that are either 3-connected or circuits or cocircuits, and the edges of T correspond to certain nested 2-separations of M . These decompositions are invariant under duality.

متن کامل

The Structure of Crossing Separations in Matroids

Oxley, Semple and Whittle described a tree decomposition for a 3-connected matroid M that displays, up to a natural equivalence, all non-trivial 3-separations of M . Crossing 3-separations gave rise to fundamental structures known as flowers. In this paper, we define a generalized flower structure called a k-flower, with no assumptions on the connectivity of M . We completely classify k-flowers...

متن کامل

The structure of 3-connected matroids of path width three

A 3-connected matroid M is sequential or has path width 3 if its ground set E(M) has a sequential ordering, that is, an ordering (e1, e2, . . . , en) such that ({e1, e2, . . . , ek}, {ek+1, ek+2, . . . , en}) is a 3-separation for all k in {3, 4, . . . , n − 3}. In this paper, we consider the possible sequential orderings that such a matroid can have. In particular, we prove that M essentially ...

متن کامل

Unavoidable minors of large 4 - connected

11 It is known that any 3-connected matroid that is large enough is certain to contain 12 a minor of a given size belonging one of a few special classes of matroids. This 13 paper proves a similar unavoidable minor result for large 4-connected bicircular 14 matroids. The main result follows from establishing the list of unavoidable minors 15 of large 4-biconnected graphs, which are the graphs r...

متن کامل

Fork-decompositions of Matroids

One of the central problems in matroid theory is Rota’s conjecture that, for all prime powers q, the class of GF (q)–representable matroids has a finite set of excluded minors. This conjecture has been settled for q ≤ 4 but remains open otherwise. Further progress towards this conjecture has been hindered by the fact that, for all q > 5, there are 3–connected GF (q)–representable matroids havin...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2011