Constructive characterizations of 3-connected matroids of path width three

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...

Fork-decompositions of ?\iatroids

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 s; 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 having...

Simpler Self-reduction Algorithm for Matroid Path-width

Path-width of matroids naturally generalizes better known path-width of graphs, and is NP-hard by a reduction from the graph case. While the term matroid path-width was formally introduced by Geelen–Gerards–Whittle [JCTB 2006] in pure matroid theory, it was soon recognized by Kashyap [SIDMA 2008] that it is the same concept as long-studied so called trellis complexity in coding theory, later na...

Constructive algorithm for path-width of matroids

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 ...