Separability Generalizes Dirac's Theorem

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Separability Generalizes Dirac's Theorem
In our study of the extremities of a graph, we define a moplex as a maximal clique module the neighborhood of which is a minimal separator of the graph. This notion enables us to strengthen Dirac’s theorem (Dirac, 1961): ‘‘Every non-clique triangulated graph has at least two non-adjacent simplicial vertices’’, restricting the definition of a simplicial vertex; this also enables us to strengthen Fulkerson and Gross’ simplicial elimination scheme; thus provides a new characterization for triangulated graphs. Our version of Dirac’s theorem generalizes from the class of triangulated graphs to any undirected graph: ‘‘Every non-clique graph has at least two non-adjacent moplexes’’. To insure a linear-time access to a moplex in any graph, we use an algorithm due to Rose Tarjan and Lueker (1976) for the recognition of triangulated graphs, known as LexBFS: we prove a new invariant for this, true even on non-triangulated graphs. 1998 Elsevier Science B.V. All rights rese...
Anne Berry, Jean Paul Bordat
Added 22 Dec 2010
Updated 22 Dec 2010
Type Journal
Year 1998
Where DAM
Authors Anne Berry, Jean Paul Bordat
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