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### Proper and unit bitolerance orders and graphs

Kenneth P. Bogart, Garth Isaak
1998 Discrete Mathematics
In this paper we show that a 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.  ...

### Tolerance Graphs [chapter]

Martin Golumbic
2013 Discrete Mathematics and Its Applications
(Remark 13.11), 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.  ...

### Bipartite tolerance orders

Kenneth P. Bogart, Ann N. Trenk
1994 Discrete Mathematics
The classes of bounded, 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  ...

### 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  ...

### Bounded bitolerance digraphs

Kenneth P. Bogart, Ann N. Trenk
2000 Discrete Mathematics
Bounded tolerance 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  we also consider directed versions of unit and proper bitolerance graphs.  ...

### Generalizations of Semiorders: A Review Note

Peter C. Fishburn
1997 Journal of Mathematical Psychology
Special thanks go to Kenneth Bogart, Ann Trenk, Garth Isaak, Jean-Paul Doignon, Jutta Mitas, 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  ...

### 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.  ...

### 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.  ...

### Author index to volume 181 (1998)

1998 Discrete Mathematics
Isaak, 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)  ...

### 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]

Breeann Flesch, Jessica Kawana, Joshua D. Laison, Dana Lapides, Stephanie Partlow, Gregory J. Puleo
2018 arXiv   pre-print
We define 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.  ...

### 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  ...

### Proper and Unit Trapezoid Orders and Graphs [article]

Kenneth P. Bogart, Rolf H. Möhring, Stephen P. Ryan
1996 arXiv   pre-print
This is different from the case of interval 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  ...

### The hierarchy of classes of bounded bitolerance orders [chapter]

Martin Charles Golumbic, Ann N. Trenk
Tolerance Graphs
We also provide separating examples between unequal classes of 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.  ...

### The Recognition of Simple-Triangle Graphs and of Linear-Interval Orders is Polynomial [article]

George B. Mertzios
2014 arXiv   pre-print
a linear 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  ), and multitolerance  ...
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