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...

• Type: Journal
• Year: 1998
• Where: DAM
• Authors: Anne Berry, Jean Paul Bordat
• Keywords: DAM 1998, Non-adjacent Simplicial Vertices, Non-clique Triangulated Graph, Triangulated Graphs