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.  

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.

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 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.

scholarly journals Power Sum Expansion of Chromatic Quasisymmetric Functions

10.37236/4761 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Christos A. Athanasiadis
Keyword(s):  
Symmetric Group ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Unit Interval ◽  
Quasisymmetric Function ◽  
Interval Order ◽  
Combinatorial Formula ◽  
Natural Unit ◽  
Power Sum ◽  
Chromatic Symmetric Function

The chromatic quasisymmetric function of a graph was introduced by Shareshian and Wachs as a refinement of Stanley's chromatic symmetric function. An explicit combinatorial formula, conjectured by Shareshian and Wachs, expressing the chromatic quasisymmetric function of the incomparability graph of a natural unit interval order in terms of power sum symmetric functions, is proven. The proof uses a formula of Roichman for the irreducible characters of the symmetric group.

scholarly journals Sobre Finura Própria de Grafos

2018 ◽  
Author(s):  
Moysés S. Sampaio Jr. ◽  
Fabiano S. Oliveira ◽  
Jayme L. Szwarcfiter
Keyword(s):  
Recognition Algorithm ◽  
Unit Interval ◽  
Interval Graphs ◽  
Graph Classes ◽  
Efficient Recognition

Both graph classes of k-thin and proper k-thin graphs have recently been introduced generalizing interval and unit interval graphs, respectively. The complexity of the recognition of k-thin and proper k-thin are open, even for fixed k 2. In this work, we introduce a subclass of the proper 2-thin graphs, called proper 2-thin of precedence. For this class, we present a characterization and an efficient recognition algorithm.

scholarly journals Minimal Obstructions for Partial Representations of Interval Graphs

10.37236/5862 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Pavel Klavik ◽  
Maria Saumell
Keyword(s):  
Linear Time ◽  
Interval Graph ◽  
Interval Graphs ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Intersection Graphs ◽  
Certifying Algorithm ◽  
Partial Representation ◽  
Partial Representation Extension ◽  
Linear Time Algorithms

Interval graphs are intersection graphs of closed intervals. A generalization of recognition called partial representation extension was introduced recently. The input gives an interval graph with a partial representation specifying some pre-drawn intervals.  We ask whether the remaining intervals can be added to create an extending representation. Two linear-time algorithms are known for solving this problem. In this paper, we characterize the minimal obstructions which make partial representations non-extendible. This generalizes Lekkerkerker and Boland's characterization of the minimal forbidden induced subgraphs of interval graphs. Each minimal obstruction consists of a forbidden induced subgraph together with at most four pre-drawn intervals. A Helly-type result follows: A partial representation is extendible if and only if every quadruple of pre-drawn intervals is extendible by itself. Our characterization leads to a linear-time certifying algorithm for partial representation extension.

FAST EXPONENTIAL-TIME ALGORITHMS FOR THE FOREST COUNTING AND THE TUTTE POLYNOMIAL COMPUTATION IN GRAPH CLASSES

2009 ◽  
Vol 20 (01) ◽  
pp. 25-44 ◽  
Author(s):  
HEIDI GEBAUER ◽  
YOSHIO OKAMOTO
Keyword(s):  
Tutte Polynomial ◽  
Unit Interval ◽  
Interval Graphs ◽  
Chordal Graphs ◽  
Regular Graphs ◽  
Exponential Time ◽  
Graph Classes ◽  
Exponential Time Algorithms ◽  
Polynomial Factor

We prove # P -completeness for counting the number of forests in regular graphs and chordal graphs. We also present algorithms for this problem, running in O *(1.8494m) time for 3-regular graphs, and O *(1.9706m) time for unit interval graphs, where m is the number of edges in the graph and O *-notation ignores a polynomial factor. The algorithms can be generalized to the Tutte polynomial computation.

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