Separability generalizes Diracʹs theorem Original Research Article

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 Diracʹs 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 Grossʹ simplicial elimination scheme; thus provides a new characterization for triangulated graphs.