Threshold tolerance graphs

In this paper, we introduce a class of graphs that generalize threshold graphs by introducing threshold tolerances. Several characterizations of these graphs are presented, one of which leads to a polynomial-time recognition algorithm. It is also shown that the complements of these graphs contain interval graphs and threshold graphs, and are contained in the subclass of chordal graphs called strongly chordal graphs, and in the class of interval tolerance graphs.

[1]  Kellogg S. Booth,et al.  Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-Tree Algorithms , 1976, J. Comput. Syst. Sci..

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

[3]  Clyde L. Monma,et al.  Tolerance graphs , 1984, Discret. Appl. Math..

[4]  F. Gavril.  The intersection graphs of subtrees in tree are exactly the chordal graphs , 1974 .

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

[6]  M. Golumbic.  Algorithmic graph theory and perfect graphs , 1980 .

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

[8]  Yechezkel Zalcstein,et al.  A Graph-Theoretic Characterization of the PV_chunk Class of Synchronizing Primitives , 1977, SIAM J. Comput..

[9]  P. Gilmore,et al.  A Characterization of Comparability Graphs and of Interval Graphs , 1964, Canadian Journal of Mathematics.

[10]  Peter L. Hammer,et al.  Threshold characterization of graphs with dilworth number two , 1985, J. Graph Theory.

[11]  Clyde L. Monma,et al.  Intersection graphs of paths in a tree , 1986, J. Comb. Theory, Ser. B.

[12]  Peter Buneman,et al.  A characterisation of rigid circuit graphs , 1974, Discret. Math..

[13]  Martin Farber,et al.  Characterizations of strongly chordal graphs , 1983, Discret. Math..

[14]  James R. Walter,et al.  Representations of chordal graphs as subtrees of a tree , 1978, J. Graph Theory.

[15]  M. Golummc.  Algorithmic graph theory and perfect graphs , 1980 .

[16]  G. Nemhauser,et al.  The k-Domination and k-Stability Problems on Sun-Free Chordal Graphs , 1984 .

[17]  P. Hammer,et al.  Aggregation of inequalities in integer programming. , 1975 .

[18]  Frank Harary,et al.  Graph Theory , 2016 .

[19]  J. Orlin.  The Minimal Integral Separator of A Threshold Graph , 1977 .

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