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.

scholarly journals Classes of Graphs with $e$-Positive Chromatic Symmetric Function

10.37236/8211 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Angèle M. Foley ◽  
Chính T. Hoàng ◽  
Owen D. Merkel
Keyword(s):  
Symmetric Functions ◽  
Interval Graph ◽  
Unit Interval ◽  
Interval Graphs ◽  
Induced Subgraphs ◽  
Related Case ◽  
Graph Classes ◽  
Unit Interval Graph ◽  
Related Graph ◽  
Chromatic Symmetric Function

In the mid-1990s, Stanley and Stembridge conjectured that the chromatic symmetric functions of claw-free co-comparability (also called incomparability) graphs were $e$-positive. The quest for the proof of this conjecture has led to an examination of other, related graph classes. In 2013 Guay-Paquet proved that if unit interval graphs are $e$-positive, that implies claw-free incomparability graphs are as well. Inspired by this approach, we consider a related case and prove  that unit interval graphs whose complement is also a unit interval graph are $e$-positive.   We introduce the concept of strongly $e$-positive to denote a graph whose induced subgraphs are all $e$-positive, and conjecture that a graph is strongly $e$-positive if and only if it is (claw, net)-free.  

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

ON THE CLIQUE-WIDTH OF SOME PERFECT GRAPH CLASSES

2000 ◽  
Vol 11 (03) ◽  
pp. 423-443 ◽  
Author(s):  
MARTIN CHARLES GOLUMBIC ◽  
UDI ROTICS
Keyword(s):  
Linear Time ◽  
Interval Graph ◽  
Unit Interval ◽  
Perfect Graphs ◽  
The Other ◽  
Perfect Graph ◽  
Permutation Graph ◽  
Permutation Graphs ◽  
Graph Classes ◽  
Unit Interval Graph

Graphs of clique–width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we study the clique–width of perfect graph classes. On one hand, we show that every distance–hereditary graph, has clique–width at most 3, and a 3–expression defining it can be obtained in linear time. On the other hand, we show that the classes of unit interval and permutation graphs are not of bounded clique–width. More precisely, we show that for every [Formula: see text] there is a unit interval graph In and a permutation graph Hn having n2 vertices, each of whose clique–width is at least n. These results allow us to see the border within the hierarchy of perfect graphs between classes whose clique–width is bounded and classes whose clique–width is unbounded. Finally we show that every n×n square grid, [Formula: see text], n ≥ 3, has clique–width exactly n+1.

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 Probe interval graphs and probe unit interval graphs on superclasses of cographs

2013 ◽  
Vol Vol. 15 no. 2 (Graph Theory) ◽  
Author(s):  
Flavia Bonomo ◽  
Guillermo Durán ◽  
Luciano N. Grippo ◽  
Martın D. Safe
Keyword(s):  
Interval Graph ◽  
Genome Project ◽  
Unit Interval ◽  
Interval Graphs ◽  
Stable Set ◽  
Induced Subgraphs ◽  
Probe Interval Graphs ◽  
Graph Class ◽  
International Audience ◽  
The Human Genome Project

Graph Theory International audience A graph is probe (unit) interval if its vertices can be partitioned into two sets: a set of probe vertices and a set of nonprobe vertices, so that the set of nonprobe vertices is a stable set and it is possible to obtain a (unit) interval graph by adding edges with both endpoints in the set of nonprobe vertices. Probe (unit) interval graphs form a superclass of (unit) interval graphs. Probe interval graphs were introduced by Zhang for an application concerning the physical mapping of DNA in the human genome project. The main results of this article are minimal forbidden induced subgraphs characterizations of probe interval and probe unit interval graphs within two superclasses of cographs: P4-tidy graphs and tree-cographs. Furthermore, we introduce the concept of graphs class with a companion which allows to describe all the minimally non-(probe G) graphs with disconnected complement for every graph class G with a companion.

New ultrastructural findings about Borrelia burgdorferi by cryofixation, negative staining, thin sectioning and scanning techniques

1990 ◽  
Vol 48 (3) ◽  
pp. 582-583
Author(s):  
S. F. Hayes ◽  
M. D. Corwin ◽  
T. G. Schwan ◽  
D. W. Dorward ◽  
W. Burgdorfer
Keyword(s):  
Lyme Disease ◽  
Borrelia Burgdorferi ◽  
Structural Characterization ◽  
Negative Staining ◽  
Low Speed ◽  
Thin Sectioning ◽  
Ultrastructural Findings

Characterization of Borrelia burgdorferi strains by means of negative staining EM has become an integral part of many studies related to the biology of the Lyme disease organism. However, relying solely upon negative staining to compare new isolates with prototype B31 or other borreliae is often unsatisfactory. To obtain more satisfactory results, we have relied upon a correlative approach encompassing a variety EM techniques, i.e., scanning for topographical features and cryotomy, negative staining and thin sectioning to provide a more complete structural characterization of B. burgdorferi.For characterization, isolates of B. burgdorferi were cultured in BSK II media from which they were removed by low speed centrifugation. The sedimented borrelia were carefully resuspended in stabilizing buffer so as to preserve their features for scanning and negative staining. Alternatively, others were prepared for conventional thin sectioning and for cryotomy using modified procedures. For thin sectioning, the fixative described by Ito, et al.

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