Betweenness and Comparability Obtained from Binary Relations

We give a brief overview of the axiomatic development of betweenness relations, and investigate the connections between these and comparability graphs. Furthermore, we characterize betweenness relations induced by reflexive and antisymmetric binary relations, thus generalizing earlier results on partial orders. We conclude with a sketch of the algorithmic aspects of recognizing induced betweenness relations.

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

[2]  Martin Charles Golumbic,et al.  Comparability graphs and a new matroid , 1977, J. Comb. Theory, Ser. B.

[3]  M. F. Smiley,et al.  Transitivities of Betweenness , 1942 .

[4]  Alfred Tarski,et al.  Tarski's System of Geometry , 1999, Bulletin of Symbolic Logic.

[5]  E. V. Huntington,et al.  A new set of postulates for betweenness, with proof of complete independence , 1924 .

[6]  R. Möhring.  Algorithmic graph theory and perfect graphs , 1986 .

[7]  M. Sholander.  Trees, lattices, order, and betweenness , 1952 .

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

[9]  Walter Wenzel,et al.  A Characterization of Ordered Sets and Lattices via Betweenness Relations , 2004 .

[10]  Martin Altwegg,et al.  Zur Axiomatik der teilweise geordneten Mengen , 1950 .

[11]  R. Möhring.  Algorithmic Aspects of Comparability Graphs and Interval Graphs , 1985 .

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

[13]  Martin Charles Golumbic,et al.  The complexity of comparability graph recognition and coloring , 1977, Computing.

[14]  M. F. Smiley,et al.  Transitives of betweenness , 1942 .

[15]  E. V. Huntington,et al.  Sets of independent postulates for betweenness , 1917 .

[16]  Bruce A. Reed,et al.  P4-comparability graphs , 1989, Discret. Math..

[17]  K. Menger.  Untersuchungen über allgemeine Metrik , 1928 .