Use this URL to cite or link to this record in EThOS:
Title: Matroids and complexity
Author: Mayhew, Dillon
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2005
Availability of Full Text:
Access from EThOS:
Full text unavailable from EThOS. Please try the link below.
Access from Institution:
We consider different ways of describing a matroid to a Turing machine by listing the members of various families of subsets, and we construct an order on these different methods of description. We show that, under this scheme, several natural matroid problems are complete in classes thought not to be equal to P. We list various results linking parameters of basis graphs to parameters of their associated matroids. For small values of k we determine which matroids have the clique number, chromatic number, or maximum degree of their basis graphs bounded above by k. If P is a class of graphs that is closed under isomorphism and induced subgraphs, then the set of matroids whose basis graphs belong to P is closed under minors. We characterise the minor-closed classes that arise in this way, and exhibit several examples. One way of choosing a basis of a matroid at random is to select a total ordering of the ground set uniformly at random and use the greedy algorithm. We consider the class of matroids having the property that this procedure chooses a basis uniformly at random. Finally we consider a problem mentioned by Oxley. He asked if, for every two elements and n - 2 cocircuits in an n-connected matroid, there is a circuit that contains both elements and that meets every cocircuit. We show that a slightly stronger property holds for regular matroids.
Supervisor: Welsh, Dominic Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Combinatorics ; matroid theory ; complexity ; basis-graph