Fragility and excluded minors

<p>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.</p>

Fragility and excluded minors

<p>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.</p>

K-Regular Matroids

<p>The class of matroids representable over all fields is the class of regular matroids. The class of matroids representable over all fields except perhaps GF(2) is the class of near-regular matroids. Let k be a non-negative integer. This thesis considers the class of k-regular matroids, a generalization of the last two classes. Indeed, the classes of regular and near-regular matroids coincide with the classes of 0-regular and 1-regular matroids, respectively. This thesis extends many results for regular and near-regular matroids. In particular, for all k, the class of k-regular matroids is precisely the class of matroids representable over a particular partial field. Every 3-connected member of the classes of either regular or near-regular matroids has a unique representability property. This thesis extends this property to the 3-connected members of the class of k-regular matroids for all k. A matroid is [omega] -regular if it is k-regular for some k. It is shown that, for all k [greater than or equal to] 0, every 3-connected k-regular matroid is uniquely representable over the partial field canonically associated with the class of [omega] -regular matroids. To prove this result, the excluded-minor characterization of the class of k-regular matroids within the class of [omega] -regular matroids is first proved. It turns out that, for all k, there are a finite number of [omega] -regular excluded minors for the class of k-regular matroids. The proofs of the last two results on k-regular matroids are closely related. The result referred to next is quite different in this regard. The thesis determines, for all r and all k, the maximum number of points that a simple rank-r k-regular matroid can have and identifies all such matroids having this number. This last result generalizes the corresponding results for regular and near-regular matroids. Some of the main results for k-regular matroids are obtained via a matroid operation that is a generalization of the operation of [Delta] - Y exchange. This operation is called segment-cosegment exchange and, like the operation of [Delta] - Y exchange, has a dual operation. This thesis defines the generalized operation and its dual, and identifies many of their attractive properties. One property in particular, is that, for a partial field P, the set of excluded minors for representability over P is closed under the operations of segment-cosegment exchange and its dual. This result generalizes the corresponding result for [Delta] - Y and Y - [Delta] exchanges. Moreover, a consequence of it is that, for a prime power q, the number of excluded minors for GF(q)-representability is at least 2q-4.</p>

K-Regular Matroids

<p>The class of matroids representable over all fields is the class of regular matroids. The class of matroids representable over all fields except perhaps GF(2) is the class of near-regular matroids. Let k be a non-negative integer. This thesis considers the class of k-regular matroids, a generalization of the last two classes. Indeed, the classes of regular and near-regular matroids coincide with the classes of 0-regular and 1-regular matroids, respectively. This thesis extends many results for regular and near-regular matroids. In particular, for all k, the class of k-regular matroids is precisely the class of matroids representable over a particular partial field. Every 3-connected member of the classes of either regular or near-regular matroids has a unique representability property. This thesis extends this property to the 3-connected members of the class of k-regular matroids for all k. A matroid is [omega] -regular if it is k-regular for some k. It is shown that, for all k [greater than or equal to] 0, every 3-connected k-regular matroid is uniquely representable over the partial field canonically associated with the class of [omega] -regular matroids. To prove this result, the excluded-minor characterization of the class of k-regular matroids within the class of [omega] -regular matroids is first proved. It turns out that, for all k, there are a finite number of [omega] -regular excluded minors for the class of k-regular matroids. The proofs of the last two results on k-regular matroids are closely related. The result referred to next is quite different in this regard. The thesis determines, for all r and all k, the maximum number of points that a simple rank-r k-regular matroid can have and identifies all such matroids having this number. This last result generalizes the corresponding results for regular and near-regular matroids. Some of the main results for k-regular matroids are obtained via a matroid operation that is a generalization of the operation of [Delta] - Y exchange. This operation is called segment-cosegment exchange and, like the operation of [Delta] - Y exchange, has a dual operation. This thesis defines the generalized operation and its dual, and identifies many of their attractive properties. One property in particular, is that, for a partial field P, the set of excluded minors for representability over P is closed under the operations of segment-cosegment exchange and its dual. This result generalizes the corresponding result for [Delta] - Y and Y - [Delta] exchanges. Moreover, a consequence of it is that, for a prime power q, the number of excluded minors for GF(q)-representability is at least 2q-4.</p>