Triangulated and Weakly Triangulated Graphs: Simpliciality in Vertices and Edges

We introduce the notion of weak simpliciality, in order to extend to weakly triangulated graphs properties of triangulated graphs, us ing Hayward’s notion that a vertexin a triangulated graph behaves as an edgein a weakly triangulated graph. In particular, we use our definition of weak simplicial edge e limination ordering to bound the number of minimal separators to n+m, and derive an efficient enumeration algorithm.

[1]  Chính T. Hoàng,et al.  Optimizing weakly triangulated graphs , 1989 .

[2]  Jeremy P. Spinrad,et al.  Weakly chordal graph algorithms via handles , 2000, SODA '00.

[3]  Anne Berry,et al.  Separability Generalizes Dirac's Theorem , 1998, Discret. Appl. Math..

[4]  Robert E. Tarjan,et al.  Algorithmic Aspects of Vertex Elimination on Graphs , 1976, SIAM J. Comput..

[5]  C. Lekkeikerker,et al.  Representation of a finite graph by a set of intervals on the real line , 1962 .

[6]  Pinar Heggernes,et al.  Recognizing Weakly Triangulated Graphs by Edge Separability , 2000, Nord. J. Comput..

[7]  Ioan Todinca,et al.  Treewidth and Minimum Fill-in of Weakly Triangulated Graphs , 1999, STACS.

[8]  Jeremy P. Spinrad,et al.  Algorithms for Weakly Triangulated Graphs , 1995, Discret. Appl. Math..

[9]  Anne Berry,et al.  Generating All the Minimal Separators of a Graph , 1999, Int. J. Found. Comput. Sci..

[10]  Ryan B. Hayward.  Meyniel Weakly Triangulated Graphs II: A Theorem of Dirac , 1997, Discret. Appl. Math..

[11]  Ryan B. Hayward.  Generating weakly triangulated graphs , 1996 .

[12]  Anne Berry,et al.  A wide-range efficient algorithm for minimal triangulation , 1999, SODA '99.

[13]  D. R. Fulkerson,et al.  Incidence matrices and interval graphs , 1965 .

[14]  Ryan B. Hayward.  Meyniel Weakly Triangulated Graphs - I: Co-perfect Orderability , 1997, Discret. Appl. Math..

[15]  D. Rose.  Triangulated graphs and the elimination process , 1970 .

[16]  A. Berry.  Désarticulation d'un graphe , 1998 .

[17]  G. Dirac.  On rigid circuit graphs , 1961 .

[18]  Ryan B. Hayward,et al.  Weakly triangulated graphs , 1985, J. Comb. Theory B.