Unit Interval Graphs of Open and Closed Intervals

2012 ◽  
Vol 72 (4) ◽  
pp. 418-429 ◽  
Author(s):  
Dieter Rautenbach ◽  
Jayme L. Szwarcfiter
Keyword(s):  
Unit Interval ◽  
Interval Graphs ◽  
Closed Intervals

scholarly journals $${\mathcal {U}}$$-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited

Algorithmica ◽  
2021 ◽  
Author(s):  
Jan Kratochvíl ◽  
Tomáš Masařík ◽  
Jana Novotná
Keyword(s):  
Unit Length ◽  
Algorithm Design ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Unit Interval ◽  
Interval Graphs ◽  
Induced Subgraph ◽  
Intersection Graphs ◽  
Proper Subclass ◽  
Maxcut Problem

AbstractInterval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs—a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a richer class of graphs. In particular, mixed unit interval graphs may contain a claw as an induced subgraph, as opposed to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyacı et al. (Inf Process Lett 121:29–33, 2017. 10.1016/j.ipl.2017.01.007). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs.

scholarly journals Multi-Armed Bandits on Partially Revealed Unit Interval Graphs

2020 ◽  
Vol 7 (3) ◽  
pp. 1453-1465 ◽  
Author(s):  
Xiao Xu ◽  
Sattar Vakili ◽  
Qing Zhao ◽  
Ananthram Swami
Keyword(s):  
Unit Interval ◽  
Interval Graphs

scholarly journals Extending Partial Representations of Proper and Unit Interval Graphs

2014 ◽  
pp. 253-264 ◽  
Author(s):  
Pavel Klavík ◽  
Jan Kratochvíl ◽  
Yota Otachi ◽  
Ignaz Rutter ◽  
Toshiki Saitoh ◽  
...  
Keyword(s):  
Unit Interval ◽  
Interval Graphs

scholarly journals ON DISTANCE TWO LABELLING OF UNIT INTERVAL GRAPHS

2009 ◽  
Vol 13 (4) ◽  
pp. 1167-1179
Author(s):  
Peter Che Bor Lam ◽  
Tao-Ming Wang ◽  
Wai Chee Shiu ◽  
Guohua Gu
Keyword(s):  
Unit Interval ◽  
Interval Graphs

scholarly journals A Characterization of Mixed Unit Interval Graphs

2014 ◽  
Vol 79 (4) ◽  
pp. 267-281 ◽  
Author(s):  
Felix Joos
Keyword(s):  
Unit Interval ◽  
Interval Graphs

Simple linear time recognition of unit interval graphs

1995 ◽  
Vol 55 (2) ◽  
pp. 99-104 ◽  
Author(s):  
Derek G Corneil ◽  
Hiryoung Kim ◽  
Sridhar Natarajan ◽  
Stephan Olariu ◽  
Alan P Sprague
Keyword(s):  
Linear Time ◽  
Unit Interval ◽  
Interval Graphs

Linear measures of some sets of the Cantor type

1957 ◽  
Vol 53 (2) ◽  
pp. 312-317 ◽  
Author(s):  
Trevor J. Mcminn
Keyword(s):  
Lebesgue Measure ◽  
Measure Zero ◽  
Cartesian Product ◽  
Open Interval ◽  
Cantor Set ◽  
Unit Interval ◽  
Linear Measure ◽  
Closed Intervals ◽  
Image Position ◽  
Dense Set

1. Introduction. Let 0 < λ < 1 and remove from the closed unit interval the open interval of length λ concentric with the unit interval. From each of the two remaining closed intervals of length ½(1 − λ) remove the concentric open interval of length ½λ(1 − λ). From each of the four remaining closed intervals of length ¼λ(1 − λ)2 remove the concentric open interval of length ¼λ(l − λ)2, etc. The remaining set is a perfect non-dense set of Lebesgue measure zero and is the Cantor set for λ = ⅓. Let Tλr be the Cartesian product of this set with the set similar to it obtained by magnifying it by a factor r > 0. Letting L be Carathéodory linear measure (1) and letting G be Gillespie linear square(2), Randolph(3) has established the following relations:

scholarly journals Completion of the mixed unit interval graphs hierarchy

2017 ◽  
Vol 87 (3) ◽  
pp. 317-332 ◽  
Author(s):  
Alexandre Talon ◽  
Jan Kratochvíl
Keyword(s):  
Unit Interval ◽  
Interval Graphs

Short Models for Unit Interval Graphs

2009 ◽  
Vol 35 ◽  
pp. 247-255 ◽  
Author(s):  
Min Chih Lin ◽  
Francisco J. Soulignac ◽  
Jayme L. Szwarcfiter
Keyword(s):  
Unit Interval ◽  
Interval Graphs

Probe interval and probe unit interval graphs on superclasses of cographs

2011 ◽  
Vol 37 ◽  
pp. 339-344
Author(s):  
Guillermo Durán ◽  
Luciano N. Grippo ◽  
Martín D. Safe
Keyword(s):  
Unit Interval ◽  
Interval Graphs
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