Fragility and excluded minors

Let Ɲ be a set of matroids. A matroid, M, is Ɲ -fragile, if for every element e, either M\e or M/e has no minor isomorphic to a member of Ɲ . This thesis gives new results in matroid representation theory that elucidate the relationship between Ɲ -fragile matroids and excluded minors. Let ℙ be a partial field, and let Ɲ be a set of strong stabilizers for ℙ. The first main result of this thesis establishes a relationship between Ɲ -fragile matroids and excluded minors for the class of ℙ-representable matroids. We prove that if an excluded minor M for the class of ℙ-representable matroids has a pair of elements a,b such that M\a,b is 3-connected with an Ɲ -minor, then either M is close to an Ɲ -minor or M\a,b is Ɲ -fragile. The result motivates a study of the structure of ℙ-representable Ɲ -fragile matroids. The matroids U₂,₅ and U₃,₅ are strong stabilizers for the U₂ and H₅ partial fields. The second main result of this thesis is a structural characterisation of the U₂- and H₅-representable {U₂,₅,U₃,₅}-fragile matroids. We prove that these matroids can be constructed from U₂,₅ and U₃,₅ by a sequence of moves, where, up to duality, each move consists of a parallel extension followed by a delta-wye or a generalised delta-wye exchange. Finally, we obtain a bound on the size of an excluded minor M for the class of U₂- or H₅-representable matroids with the property that M has a pair of elements a,b such that M\a,b is 3-connected with a {U₂,₅,U₃,₅}-minor. Our proof uses the first and second main results of this thesis