Darren B. Parker

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In the context of two-path convexity, we study the rank, Helly number, Radon number, Caratheodory number, and hull number for multipartite tournaments. We show the maximum Caratheodory number of a multipartite tournament is 3. We then derive tight upper bounds for rank in both general multipartite tournaments and clone-free multipartite tournaments. We show(More)
We present some results on two-path convexity in clone-free regular multipartite tournaments. After proving a structural result for regular multipartite tournaments with convexly independent sets of a given size, we determine tight upper bounds for their size (called the rank) in clone-free regular bipartite and tripartite tournaments. We use this to(More)
We study two-path convexity in bipartite tournaments. For a bipartite tournament , we obtain both a necessary condition and a sufficient condition on the adjacency matrix for its rank to be two. We then investigate 4-cycles in bipartite tournaments of small rank. We show that every vertex in a bipartite tournament of rank two lies on a four cycle, and(More)
We investigate the convex invariants associated with two-path convexity in clone-free multipartite tournaments. Specifically, we explore the relationship between the Helly number, Radon number and rank of such digraphs. The main result is a structural theorem that describes the arc relationships among certain vertices associated with vertices of a given(More)
The collection of convex subsets of a multipartite tournament T forms a lattice C(T). Given a lattice structure for C(T), we deduce properties of T. In particular, we find conditions under which we can detect clones in T. We also determine conditions on the lattice which will imply that T is bipartite, except for a few cases. We classify the ambiguous(More)
We call T = (G1, G2, G3) a graph-triple of order t if the Gi are pairwise non-isomorphic graphs on t non-isolated vertices whose edges can be combined to form Kt. If m ≥ t, we say T divides Km if E(Km) can be partitioned into copies of the graphs in T with each Gi used at least once, and we call such a partition a T-multidecomposition. In this paper, we(More)
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