Separability Generalizes Dirac's Theorem

نویسندگان

Anne Berry

Jean Paul Bordat

چکیده

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 reserved.

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