# Maximal independent sets and maximal matchings in series-parallel and related graph classes

## TL;DR

This paper analyzes the number and size of maximal independent sets and matchings in series-parallel and related graph classes using generating functions and singularity analysis, extending previous results for trees.

## Contribution

It provides quantitative results for block-stable graph classes under sub-criticality, applying advanced generating function techniques to extend known tree results.

## Key findings

- Quantitative bounds on maximal independent sets and matchings
- Extension of Meir and Moon's results to broader graph classes
- Methodology applicable to various block-stable graph classes

## Abstract

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].