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Proper and unit bitolerance orders and graphs

1998
*
Discrete Mathematics
*

In this paper we show that a

doi:10.1016/s0012-365x(97)00043-5
fatcat:umugguvcdvam5o4ye7bhppe3da
*bitolerance**graph*or*order*is*proper*if*and*only if it is*unit*. ... Such a*graph*or*order*is called*proper*if it has a representation using intervals no one of which is a*proper*subset of another,*and*it is called*unit*if it has a representation using only*unit*intervals ... It is clear that a*unit*interval*order*(*graph*) is*proper*. Roberts first noted that a*proper*interval*graph*(*order*) is*unit*. ...##
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Tolerance Graphs
[chapter]

2013
*
Discrete Mathematics and Its Applications
*

(Remark 13.11),

doi:10.1201/b16132-66
fatcat:j2i3oyr2e5au7mtjsdmhmcaezm
*unit*,*proper*,*and*point-core*bitolerance*digraphs (Corollary 13.41). ... Trees, Cotrees,*and*Bipartite*graphs*11. Tolerance Models on Trees 4. Interval Probe*Graphs*5.*Bitolerance**and**Ordered*Sets 6.*Unit**and*50% Tolerance*Orders*7. ...##
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Bipartite tolerance orders

1994
*
Discrete Mathematics
*

The classes of bounded,

doi:10.1016/0012-365x(92)00571-8
fatcat:as5pfz27ubfxhhbpbljjo7ybnm
*proper**and**unit*tolerance*orders**and**graphs*have been studied by other authors. Here we introduce two-sided versions of these classes. ... Tolerance*orders**and*tolerance*graphs*arise as a generalization of interval*orders**and*interval*graphs*in which some overlap of intervals is tolerated. ... Then the following are equivalent: (1) P is a*unit*tolerance*order*, (2) P is a 50% tolerance*order*, (3) P is a*unit**bitolerance**order*, (4) P is a*proper*tolerance*order*, (5) P is a*proper**bitolerance**order*...##
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Page 7558 of Mathematical Reviews Vol. , Issue 98M
[page]

1998
*
Mathematical Reviews
*

The authors show that the class of

*unit**bitolerance*digraphs*and*the class of*proper**bitolerance*digraphs are equal,*and*that they are the same as the class of interval catch digraphs. ... A*bitolerance*digraph is called*proper*if it has a representation in which no interval is a*proper*subset of another one,*and**unit*if it has a representation in which all intervals have the same length ...##
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Bounded bitolerance digraphs

2000
*
Discrete Mathematics
*

Bounded tolerance

doi:10.1016/s0012-365x(99)00220-4
fatcat:ckboonabdzd63d5o3cmtvlvwru
*graphs*were introduced in Golumbic*and*Monma (Congr. Numer. 35 (1982) 321-331)*and*Golumbic et al. (Discrete Appl. Math. 9 (1984) 157-170) as a generalization of interval*graphs*. ... In addition, we characterize those bounded*bitolerance*digraphs which arise when linear*orders*or weak*orders*are used in place of interval*orders*. ... In [2] we also consider directed versions of*unit**and**proper**bitolerance**graphs*. ...##
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Generalizations of Semiorders: A Review Note

1997
*
Journal of Mathematical Psychology
*

Special thanks go to Kenneth Bogart, Ann Trenk, Garth Isaak, Jean-Paul Doignon, Jutta Mitas,

doi:10.1006/jmps.1997.1179
pmid:9473398
fatcat:6sons3jamncyloz4x656scuqp4
*and*Marc Pirlot. ... ACKNOWLEDGMENTS I am indebted to several people for their careful reading of the original manuscript*and*their suggestions for improvements. ... The classes of*bitolerance**orders**and*trapezoid*orders*are identical, as are the classes of split interval*orders**and**unit**bitolerance**orders*, the classes of*unit**bitolerance**orders**and**proper**bitolerance*...##
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Page 8509 of Mathematical Reviews Vol. , Issue 2004k
[page]

2004
*
Mathematical Reviews
*

Requirements on intervals may be (1) equal length (

*unit*bitoler- ance*orders*), (2) avoidance of*proper*containment (*proper*bitol- erance*orders*), or (3) no restriction. ... Several of the equivalences are of the “*proper*=*unit*” variety, though this does not always hold. Also class 3aii is equivalent to classes 2cii*and*Icii,*and*classes 3ai*and*Ici are equivalent. ...##
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Page 5126 of Mathematical Reviews Vol. , Issue 95i
[page]

1995
*
Mathematical Reviews
*

Further concepts are defined:

*unit*tolerance*order*, 50% tolerance*order*,*unit**bitolerance**order*,*proper*tolerance*order*,*proper**bitolerance**order*, totally bounded tolerance*order*, totally bounded*bitolerance*... (D-TUB; Berlin) Treewidth of cocomparability*graphs**and*a new*order*-theoretic parameter. (English summary)*Order*11 (1994), no. 1, 47-60. ...##
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Author index to volume 181 (1998)

1998
*
Discrete Mathematics
*

Isaak,

doi:10.1016/s0012-365x(97)82062-6
fatcat:sqdp43rjovam5az5gxnwxknhxq
*Proper**and**unit**bitolerance**orders**and**graphs*(1-3 Bouchet, A.*and*W. Schw~irzler, The delta-sum of matching delta-matroids (1-3 Brandt, S., see D. Amar , M.*and*A.V. ... Schoen, Independent finite sums in*graphs*defined on the natural numbers (Note) , A.N., On k-weak*orders*: Recognition*and*a tolerance result (1-3) Tsai, Y.S., see H.J. Shyr (1-3) ...##
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Page 5194 of Mathematical Reviews Vol. , Issue 99h
[page]

1999
*
Mathematical Reviews
*

Vijayakumar (Cochin)
99h:05098 05C75
Bogart, Kenneth P. (1-DTM-CS; Hanover, NH);
Isaak, Garth (1-DTM-CS; Hanover, NH)

*Proper**and**unit**bitolerance**orders**and**graphs*. ... The comparability*graphs*of these*ordered*sets are then considered. Characterizations are given in the case of bounded tolerances,*proper*tolerance*orders*,*and**unit*tolerance*orders*. ...##
###
Veto Interval Graphs and Variations
[article]

2018
*
arXiv
*
pre-print

We define

arXiv:1709.09259v2
fatcat:5h2zjm5vyfggngsvy2uky7bpke
*and*prove similar results about several related*graph*families, including*unit*VI*graphs*, midpoint*unit*VI (MUVI)*graphs*,*and*single*and*double approval*graphs*. ... We also highlight a relationship between approval*graphs**and*a family of tolerance*graphs*. ... Acknowledgements We thank Benjamin Reiniger for suggesting the House of*Graphs*as a source of large chromatic number triangle-free*graphs*. ...##
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Page 6776 of Mathematical Reviews Vol. , Issue 98K
[page]

1998
*
Mathematical Reviews
*

Nine nonequivalent classes are designated as interval

*orders*, bisemiorders, split semiorders,*unit*tolerance*orders*,*unit**bitolerance**orders*, tolerance*orders*,*bitolerance*or- ders, semitransitive*orders*... As corollaries we deduce Dilworth’s theorem*and*the well- known min-max formula for the minimum size edge cover of a*graph*.” 98k:06006 06A07 KlaSka, Jifi (CZ-TUB; Brno) Partitions*and*partially*ordered*...##
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Proper and Unit Trapezoid Orders and Graphs
[article]

1996
*
arXiv
*
pre-print

This is different from the case of interval

arXiv:math/9611215v1
fatcat:wh3latyq7nclrelwmixbu34f5y
*orders*, where the class of*proper*interval*orders*is exactly the same as the class of*unit*interval*orders*. ... We show that the class of trapezoid*orders*in which no trapezoid strictly contains any other trapezoid strictly contains the class of trapezoid*orders*in which every trapezoid can be drawn with*unit*area ... The class of trapezoid*orders*(*graphs*) is also the same as the class of bounded*bitolerance**orders*(*graphs*), though the natural definitions of*proper**and**unit*are different in that context [Langley, 1993 ...##
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The hierarchy of classes of bounded bitolerance orders
[chapter]

*
Tolerance Graphs
*

We also provide separating examples between unequal classes of

doi:10.1017/cbo9780511542985.012
fatcat:3zfd7cagdnfmdflp5xbxlenlom
*bitolerance**orders*. Definition 2. (*Unit*): P is a*unit**bitolerance**order*if it has a bounded*bitolerance*representation Definition 3. ... (*Proper*): P is a*proper**bitolerance**order*if it has a bounded*bitolerance*representation I, p, q in which I x ⊂ I y for all x, y ∈ V . Restrictions on Tolerant Points p(v), q(v) Definition 4. ... The classes of*unit*totally bounded*bitolerance**orders*(1bii)*and**proper*totally bounded*bitolerance**orders*(2bii) are equivalent.Proof. ...##
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The Recognition of Simple-Triangle Graphs and of Linear-Interval Orders is Polynomial
[article]

2014
*
arXiv
*
pre-print

a linear

arXiv:1210.4352v2
fatcat:hrmumubukncsne23kdsrqji5ye
*order**and*P_2 is an interval*order*. ... They lie naturally between permutation*and*trapezoid*graphs*, which are the intersection*graphs*of line segments between L_1*and*L_2*and*of trapezoids between L_1*and*L_2, respectively. ... (these are subclasses of parallelogram*graphs*,*and*thus also subclasses of trapezoid*graphs*),*proper**bitolerance**graphs*[2, 11] (they coincide with*unit**bitolerance**graphs*[2] ),*and*multitolerance ...
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