scholarly journals The Roberts characterization of proper and unit interval graphs

2007 ◽  
Vol 307 (22) ◽  
pp. 2906-2908 ◽  
Author(s):  
Frédéric Gardi
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

scholarly journals A Structural Characterization for Certifying Robinsonian Matrices

10.37236/6701 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Monique Laurent ◽  
Matteo Seminaroti ◽  
Shin-ichi Tanigawa
Keyword(s):  
Structural Characterization ◽  
Adjacency Matrix ◽  
Efficient Algorithm ◽  
Symmetric Matrix ◽  
Interval Graph ◽  
Unit Interval ◽  
Interval Graphs ◽  
Weighted Graphs ◽  
Unit Interval Graph

A symmetric matrix is Robinsonian if its rows and columns can be simultaneously reordered in such a way that entries are monotone nondecreasing in rows and columns when moving toward the diagonal. The adjacency matrix of a graph is Robinsonian precisely when the graph is a unit interval graph, so that Robinsonian matrices form a matrix analogue of the class of unit interval graphs. Here we provide a structural characterization for Robinsonian matrices in terms of forbidden substructures, extending the notion of  asteroidal triples to weighted graphs. This implies the known characterization of unit interval graphs and leads to an efficient algorithm for certifying that a matrix is not Robinsonian.

Recognition and characterization of unit interval graphs with integer endpoints

2018 ◽  
Vol 245 ◽  
pp. 168-176
Author(s):  
G. Durán ◽  
F. Fernández Slezak ◽  
L.N. Grippo ◽  
F.de S. Oliveira ◽  
J.L. Szwarcfiter
Keyword(s):  
Unit Interval ◽  
Interval Graphs

scholarly journals Classes of graphs with restricted interval models

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Andrzej Proskurowski ◽  
Jan Arne Telle
Keyword(s):  
Maximum Clique ◽  
Interval Graphs ◽  
Induced Subgraph ◽  
Graph Theoretic ◽  
Graph Classes ◽  
International Audience ◽  
Forbidden Induced Subgraph ◽  
Nested Hierarchy ◽  
Graph Families

International audience We introduce q-proper interval graphs as interval graphs with interval models in which no interval is properly contained in more than q other intervals, and also provide a forbidden induced subgraph characterization of this class of graphs. We initiate a graph-theoretic study of subgraphs of q-proper interval graphs with maximum clique size k+1 and give an equivalent characterization of these graphs by restricted path-decomposition. By allowing the parameter q to vary from 0 to k, we obtain a nested hierarchy of graph families, from graphs of bandwidth at most k to graphs of pathwidth at most k. Allowing both parameters to vary, we have an infinite lattice of graph classes ordered by containment.

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

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
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