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Vertices x and y dominate a tournament T if for all vertices z 6 = x; y; either x beats z or y beats z. Let dom(T) be the graph on the vertices of T with edges between pairs of vertices that dominate T. We show dom(T) is either an odd cycle with possible pendant vertices or a forest of caterpillars. Since dom(T) is the complement of the competition graph of… (More)

The competition graph of a loopless symmetric digraph If is the rwo-.\rc'p grclph. S,(H). Necessary and sufficient conditions on If are given for S,(ff) to be interval or unit interval. These are useful properties when application requires that the competition graph be efficiently colorable. Computational aspects are discussed. as are related open problems.… (More)

- R Lundgren, O Lib-Man, J Totten, K Weaver, Peidong Yang, Tao Deng +6 others
- 1998

Porous silica, niobia, and titania with three-dimensional structures patterned over multiple length scales were prepared by combining micromolding, polystyrene sphere templating, and cooperative assembly of inorganic sol-gel species with amphiphilic triblock copolymers. The resulting materials show hierarchical ordering over several discrete and tunable… (More)

We study the minimum number of complete bipartite subgraphs needed to cover and partition the edges of a k-regular bigraph on 2n vertices. Bounds are determined on the minima of these numbers for fixed n and k. Exact values of the minima are found for all n and k 6 4. The same results hold for directed graphs. Equivalently, we have determined bounds on the… (More)

If D is an acyclic digraph, its competition graph has the same vertex set as D and an edge between vertices x and y if and only if for some vertex u, there are arcs (_q u) and (_Y, u) in D. We study competition graphs of acyclic digraphs D when the indegrees and outdegrees of the vertices of D are restricted. Under degree restrictions, we characterize the… (More)