scholarly journals On the Maximum Cardinality Cut Problem in Proper Interval Graphs and Related Graph Classes

2021 ◽  
Author(s):  
Arman Boyacı ◽  
Tınaz Ekim ◽  
Mordechai Shalom
Keyword(s):  
Interval Graphs ◽  
Maximum Cardinality ◽  
Graph Classes ◽  
Related Graph

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 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.

-domination for chordal graphs and related graph classes

2013 ◽  
Vol 44 ◽  
pp. 219-224 ◽  
Author(s):  
Gabriela Argiroffo ◽  
Valeria Leoni ◽  
Pablo Torres
Keyword(s):  
Chordal Graphs ◽  
Graph Classes ◽  
Related Graph

scholarly journals Certain chromatic sums of some cycle-related graph classes

2016 ◽  
Vol 08 (03) ◽  
pp. 1650050 ◽  
Author(s):  
N. K. Sudev ◽  
K. P. Chithra ◽  
Johan Kok
Keyword(s):  
Graph Classes ◽  
Related Graph ◽  
Sum Formula ◽  
Chromatic Sum

Let [Formula: see text] be a certain type of proper [Formula: see text]-coloring of a given graph [Formula: see text] and [Formula: see text] denote the number of times a particular color [Formula: see text] is assigned to the vertices of [Formula: see text]. Then, the coloring sum of a given graph [Formula: see text] with respect to the coloring [Formula: see text], denoted by [Formula: see text] is defined to be [Formula: see text]. The coloring sums such as [Formula: see text]-chromatic sum, [Formula: see text]-chromatic sum, [Formula: see text]-chromatic sum, [Formula: see text]-chromatic sum, etc. are some of these types of coloring sums that have been studied recently. Motivated by these studies on certain chromatic sums of graphs, in this paper, we study certain chromatic sums for some standard cycle-related graphs.

The 2-dominating set polytope of cycles and related graph classes

2013 ◽  
Vol 44 ◽  
pp. 269-274
Author(s):  
G. Argiroffo ◽  
M. Escalante ◽  
M.E. Ugarte
Keyword(s):  
Dominating Set ◽  
Graph Classes ◽  
Related Graph

scholarly journals Isomorphism of graph classes related to the circular-ones property

2013 ◽  
Vol Vol. 15 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Andrew R. Curtis ◽  
Min Chih Lin ◽  
Ross M. Mcconnell ◽  
Yahav Nussbaum ◽  
Francisco Juan Soulignac ◽  
...  
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Circular Arc ◽  
Graph Classes ◽  
Discrete Algorithms ◽  
International Audience ◽  
Related Graph ◽  
Circular Arc Graphs ◽  
Proper Circular Arc Graphs

Discrete Algorithms International audience We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. Our algorithm is similar to the isomorphism algorithm for interval graphs of Lueker and Booth, but works on PC trees, which are unrooted and have a cyclic nature, rather than with PQ trees, which are rooted. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc graphs, Γ circular-arc graphs, proper circular-arc graphs and convex-round graphs.

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 Maximal Independent Sets and Maximal Matchings in Series-Parallel and Related Graph Classes

10.37236/8683 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Michael Drmota ◽  
Lander Ramos ◽  
Clément Requilé ◽  
Juanjo Rué
Keyword(s):  
Functional Equations ◽  
Independent Sets ◽  
Singularity Analysis ◽  
Graph Classes ◽  
Several Variables ◽  
In Series ◽  
Quantitative Results ◽  
Related Graph ◽  
Implicit Systems ◽  
Maximal Independent Sets

The goal of this paper is to obtain quantitative results on the number and on the size of maximal independent sets and maximal matchings in several block-stable graph classes that satisfy a proper sub-criticality condition. In particular we cover trees, cacti graphs and series-parallel graphs. The proof methods are based on a generating function approach and a proper singularity analysis of solutions of implicit systems of functional equations in several variables. As a byproduct, this method extends previous results of Meir and Moon for trees [Meir, Moon: On maximal independent sets of nodes in trees, Journal of Graph Theory 1988].

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