Simplicial Vertices Research Articles

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.

A Characterization of Some Graph Classes with No Long Holes

We give a characterization of a hierarchy of graph classes with no long holes in which each class excludes some long antiholes. Al one end of the hierarchy is the class of graphs with no long holes. At the other end is the class of weakly triangulated graphs. The characterization has the flavor of that for triangulated graphs: G is a triangulated graph if and only if every induced subgraph H of G is either a clique or has two nonadjacent simplicial vertices of H.

On the tree representation of chordal graphs

AbstractWe introduce the notion of the boundary clique and the k‐overlap clique graph and prove the following: Every incomplete chordal graph has two nonadjacent simplicial vertices lying in boundary cliques. An incomplete chordal graph G is k‐connected if and only if the k‐overlap clique graph gk(G) is connected. We give an algorithm to construct a clique tree of a connected chordal graph and characterize clique trees of connected chordal graphs using the algorithm.