Characterization and representation problems for intersection betweennesses

For a set system M=(M"v)"v"@?"V indexed by the elements of a finite set V, the intersection betweenness B(M) induced by M consists of all triples (u,v,w)@?V^3 with M"[email protected]?M"[email protected]?M"v. Similarly, the strict intersection betweenness B"s(M) induced by M consists of all triples (u,v,w)@?B(M) such that u, v, and w are pairwise distinct. The notion of a strict intersection betweenness was introduced by Burigana [L. Burigana, Tree representations of betweenness relations defined by intersection and inclusion, Math. Soc. Sci. 185 (2009) 5-36]. We provide axiomatic characterizations of intersection betweennesses and strict intersection betweennesses. Our results yield a simple and efficient algorithm that constructs a representing set system for a given (strict) intersection betweenness. We study graphs whose strict shortest path betweenness is a strict intersection betweenness. Finally, we explain how the algorithmic problem related to Burigana's notion of a partial tree representation can be solved efficiently using well-known algorithms.

[1]  M. Golumbic.  Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57) , 2004 .

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

[3]  Yury Makarychev.  Simple linear time approximation algorithm for betweenness , 2012, Oper. Res. Lett..

[4]  Dieter Rautenbach,et al.  Finite Sholander trees, trees, and their betweenness , 2011, Discret. Math..

[5]  Nellie Clarke Brown.  Trees , 1896, Savage Dreams.

[6]  R. Myers,et al.  Radiation hybrid mapping: a somatic cell genetic method for constructing high-resolution maps of mammalian chromosomes. , 1990, Science.

[7]  Madhu Sudan,et al.  A Geometric Approach to Betweenness , 1995, ESA.

[8]  Noga Alon,et al.  Ordinal embeddings of minimum relaxation: general properties, trees, and ultrametrics , 2005, SODA '05.

[9]  Gregory Gutin,et al.  Ordinal Embedding Relaxations Parameterized Above Tight Lower Bound , 2009, ArXiv.

[10]  Dieter Rautenbach,et al.  Finite Sholander Trees , Trees , and their Betweennesses , 2009 .

[11]  Peter Jeavons,et al.  Constraint Satisfaction Problems on Intervals and Length , 2004, SIAM J. Discret. Math..

[12]  Clement T. Yu,et al.  An algorithm for tree-query membership of a distributed query , 1979, COMPSAC.

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

[14]  S. Goss,et al.  New method for mapping genes in human chromosomes , 1975, Nature.

[15]  F. McMorris,et al.  Topics in Intersection Graph Theory , 1987 .

[16]  Luigi Burigana,et al.  TREE REPRESENTATIONS OF BETWEENNESS RELATIONS DEFINED BY INTERSECTION AND INCLUSION , 2009 .

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

[18]  Jaroslav Opatrny,et al.  Total Ordering Problem , 1979, SIAM J. Comput..

[19]  Venkatesan Guruswami,et al.  Every Permutation CSP of arity 3 is Approximation Resistant , 2009, 2009 24th Annual IEEE Conference on Computational Complexity.

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

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

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

[23]  D. Defays,et al.  Tree representations of ternary relations , 1979 .

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