Feodor Dragan

1 I n t r o d u c t i o n All graphs in this paper are connected and simple, i.e. finite, undirected, loopless and without multiple edges. In a graph G = (V, E) the length of a path from a vertex v to a vertex u is the number of edges in the path. The distance d(u, v) from vertex u to vertex v is the length of a minimum length path from u to v and the interval I(u, v) between these vertices is the set i(u, v) = {w e v: d(u, v) = a(u, w) + a(~, v)}. The eccentricity e(v) of a vertex v is the maximum distance from v to any vertex in G. Denote by D(v) the set of all farthest from v vertices, i.e. O(v) = {w r V : d(v, w) = e(v)}. The radius r(G) is the minimum eccentricity of a vertex in G and the diameter d(G) the maximum eccentricity. The center C(G) of G is the subgraph induced by the set of all central vertices , i.e. vertices whose eccentricities are equal to r(G). A clique of G is a set of pairwise adjacent vertices. A vertex v of G is called simplicial if its neighborhood is a clique. A graph G is chordal (triangulated) if every cycle of length greater than three possesses a chord, i.e. an edge joining two nonconsecutive vertices on the cycle. Chordal graphs arise in the study of Gaussian elimination of sparse symmetric *This work was partially supported by the VW-Stiftung Project No. 1/69041 e-mail addresses: [email protected], [email protected]