Showing papers on "K-tree published in 1991"
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TL;DR: It is shown that deciding membership in a correlation polytope is an NP-complete problem, and deciding facets is probably not even in NP.
Abstract: A family of polytopes, correlation polytopes, which arise naturally in the theory of probability and propositional logic, is defined. These polytopes are tightly connected to combinatorial problems in the foundations of quantum mechanics, and to the Ising spin model. Correlation polytopes exhibit a great deal of symmetry. Exponential size symmetry groups, which leave the polytope invariant and act transitively on its vertices, are defined. Using the symmetries, a large family of facets is determined. A conjecture concerning the full facet structure of correlation polytopes is formulated (the conjecture, however, implies that NP=co-NP). Various complexity results are proved. It is shown that deciding membership in a correlation polytope is an NP-complete problem, and deciding facets is probably not even in NP. The relations between the polytope symmetries and its complexity are indicated.
190 citations
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TL;DR: The following conjecture of T. Gallai is proved: If G is a chordal graph on n vertices, such that all its maximal complete subgraphs have order at least 3, then there is a vertex set of cardinality ⩽n 3 which meets all maximal completeSubgraphs of G.
135 citations
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TL;DR: The threshold for the existence of a spanning maximal planar subgraph in the random graph Gn, p is studied and it is shown that it is very near p = 1/n⅓.
Abstract: We study the threshold for the existence of a spanning maximal planar subgraph in the random graph Gn, p . We show that it is very near p = 1/n⅓ We also discuss the threshold for the existence of a spanning maximal outerplanar subgraph. This is very near p = 1/n½.
6 citations
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TL;DR: In this article, the authors use the theory of coherent configurations to establish the spectra (and hence the dimension) of classes of faces induced by other substructures (e.g., cliques) of the structure.
1 citations