Research Information for Darren Parker

Professional Information

Games on Graphs, Convexity in Directed Graphs, Coalgebras, Hopf Algebras, and whatever else gets thrown my way.

J. Jones, D.B. Parker, and V. Zadorozhnyy, A Group Labeling Version of the Lights Out Game, submitted to Involve.

D.B. Parker, The Lights-Out Game on Subdivided Caterpillars, (to appear in Ars Combinatoria).

A. Giffen and D.B. Parker, On Generalizing the ``Lights Out'' Game and a Generalization of Parity Domination, Ars Combinatoria 111 (2013), pp. 273--288.

D.B. Parker, R.F. Westhoff, and M.J. Wolf, Two-Path Convexity and Bipartite Tournaments of Small Rank, Ars Combinatoria 97 (2010), pp. 181--191.

D.B. Parker, R.F. Westhoff, and M.J. Wolf, Convex Independence and the Structure of Clone-Free Multipartite Tournaments, Discussiones Mathematicae Graph Theory 29(1) (2009), pp. 51-69

D.B. Parker, R.F. Westhoff, and M.J. Wolf, On Two-Path Convexity in Multipartite Tournaments, European Journal of Combinatorics 29 (2008), pp. 641-651.

A. Abueida, M. Daven, W.S. Diestelkamp, S.P. Edwards, and D.B. Parker, Multidesigns for Graph-Triples of Order 6, Congressus Numerantium 183 (2006), pp. 139-160.

D.B. Parker, R.F. Westhoff, and M.J. Wolf, Two-Path Convexity in Clone-Free Regular Multipartite Tournaments, Australasian Journal of Combinatorics 36 (2006), pp. 177-196..

A. Abueida, W.S. Diestelkamp, S.P. Edwards, and D.B. Parker, Determining Properties of a Multipartite Tournament from its Lattice of Convex Subsets, Australasian Journal of Combinatorics 31 (2005), pp. 217-230.

D.B. Parker, On the Coradical Filtration of Pointed Coalgebras, Journal of Algebra 255 (2002), pp. 121-134.

D.B. Parker, U(g)-Galois Extensions, Communications in Algebra 29 (2001), pp. 2859-2870.

D.B. Parker, Forms of Coalgebras and Hopf Algebras, Journal of Algebra 239 (2001), pp. 1-34.

D.B. Parker, Hopf Galois Extensions and Forms of Coalgebras and Hopf Algebras, Doctoral Thesis.

With L. Keough: We are working on an extremal problem in the Lights Out (POLO) game. On complete graphs K_n with n at least 2, it is easy to label the vertices in a way that makes the game impossible to win. We seek to determine all graphs of order n and of maximum size such that the game can be won regardless of the initial labeling.
With R. Hutchings, S. Napier, and L. Renaud: We are looking for which graphs in given graph classes (wheel graphs, complete multipartite graphs, and certain directed graphs) for which the POLO game can be won regardless of the initial labeling.
With R.F. Westhoff and M.J. Wolf: We are studying multipartite tournaments and their convex subsets under two-path convexity. In particular, we are studying a convexity invariant called rank. Rank is an upper bound for the Helly number, the Caratheodory number, and the Radon number, the three most central numbers in abstract convexity theory. In addition, it is an upper bound for the hull number, as well as the number of vertices required to generate all convex subsets using convex hulls. In particular, we are interested in finding classes of digraphs for which the Helly number, Radon number, and rank are all equal.

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