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].

Trees without twin-leaves with smallest number of maximal independent sets

2020 ◽  
Vol 30 (1) ◽  
pp. 53-67 ◽  
Author(s):  
Dmitriy S. Taletskii ◽  
Dmitriy S. Malyshev
Keyword(s):  
Independent Sets ◽  
Adjacent Vertex ◽  
Maximal Independent Sets

AbstractFor any n, in the set of n-vertex trees such that any two leaves have no common adjacent vertex, we describe the trees with the smallest number of maximal independent sets.

On graphs with the third largest number of maximal independent sets

2009 ◽  
Vol 109 (4) ◽  
pp. 248-253 ◽  
Author(s):  
Hongbo Hua ◽  
Yaoping Hou
Keyword(s):  
Independent Sets ◽  
The Third ◽  
Maximal Independent Sets

scholarly journals Maximal independent sets on a grid graph

2017 ◽  
Vol 340 (12) ◽  
pp. 2762-2768 ◽  
Author(s):  
Seungsang Oh
Keyword(s):  
Independent Sets ◽  
Grid Graph ◽  
Maximal Independent Sets

Functional Equations in Several Variables

2020 ◽  
pp. 20-57
Author(s):  
Jaroslav Smίtal
Keyword(s):  
Functional Equations ◽  
Several Variables

Combinatorial aspects of the Löwenstein avoidance rule. Part I: the independence polynomial

2021 ◽  
Vol 77 (6) ◽  
Author(s):  
Montauban Moreira de Oliveira Jr ◽  
Jean-Guillaume Eon
Keyword(s):  
Propositional Calculus ◽  
Independent Sets ◽  
Companion Paper ◽  
Theoretical Interpretation ◽  
Vector Method ◽  
Independence Polynomial ◽  
Calculation Technique ◽  
Independence Polynomials ◽  
Maximal Independent Sets

According to Löwenstein's rule, Al–O–Al bridges are forbidden in the aluminosilicate framework of zeolites. A graph-theoretical interpretation of the rule, based on the concept of independent sets, was proposed earlier. It was shown that one can apply the vector method to the associated periodic net and define a maximal Al/(Al+Si) ratio for any aluminosilicate framework following the rule; this ratio was called the independence ratio of the net. According to this method, the determination of the independence ratio of a periodic net requires finding a subgroup of the translation group of the net for which the quotient graph and a fundamental transversal have the same independence ratio. This article and a companion paper deal with practical issues regarding the calculation of the independence ratio of mainly 2-periodic nets and the determination of site distributions realizing this ratio. The first paper describes a calculation technique based on propositional calculus and introduces a multivariate polynomial, called the independence polynomial. This polynomial can be calculated in an automatic way and provides the list of all maximal independent sets of the graph, hence also the value of its independence ratio. Some properties of this polynomial are discussed; the independence polynomials of some simple graphs, such as short paths or cycles, are determined as examples of calculation techniques. The method is also applied to the determination of the independence ratio of the 2-periodic net dhc.

Local Structure When All Maximal Independent Sets Have Equal Weight

1998 ◽  
Vol 11 (4) ◽  
pp. 644-654 ◽  
Author(s):  
Yair Caro ◽  
M. N. Ellingham ◽  
J. E. Ramey
Keyword(s):  
Local Structure ◽  
Independent Sets ◽  
Equal Weight ◽  
Maximal Independent Sets
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