Clique-width and well-quasi-order Case for Support Previous Track Record

Previous Track Record Robert Brignall (PI) has been a Lecturer in Combinatorics at The Open University since 2010. He received his PhD in 2007 from the University of St Andrews, and from 2007–2010 he was a Heilbronn Research Fellow at The University of Bristol. In Bristol, he spent 50% of his time on classified research directed by the Heilbronn Institute, and 50% on his own research agenda. He supervises one PhD student who is on track to finish in the first half of 2015 (within three years of the start of his PhD), and he supervised a Research Assistant from 2012–14, part-funded by EPSRC grant EP/J006130/1, who now holds a permanent academic position in the UK. He has written 17 peer-reviewed papers, with a further 2 currently under review. His career began in the structural study of permutation classes, and he continues to make lasting contributions to this area [10, 12, 13, 16, 18], and its consequences for the enumeration of permutation classes [1–4]. Following his PhD, his work expanded in two directions: first, he looked at the question of well-quasi-ordering for permutations, and this resulted in a single-author paper which represents the state-of-the-art in infinite antichain construction [11]. Second, he applied his structural expertise to the wider study of combinatorial structures [17], with a particular emphasis on the cross-fertilisation of results between permutations and graphs. His research in these two directions were combined when in 2012 he became PI on grant EP/J006130/1. Through new collaborations with researchers in Warwick (including the named RA on this proposal), this grant catalysed the process of translating structural results from permutations to graphs [15], most especially with regards to well-quasi-ordering and infinite antichains [7]. The importance to the study of structure and well-quasi-ordering in graphs of the second subject of this proposal, clique-width, was brought to the attention of the PI during the course of this grant. Consequently, the PI undertook to enhance his intuition of this area, particularly in identifying the interface between graph classes where clique-width is bounded, and graph classes where clique-width is unbounded [6]. He also considered similar questions for a restricted version of this parameter called linear clique-width [14], which is important both to extend our understanding of the more general problem, and because it has direct relevance to the construction of infinite antichains.