On the saturation number of graphs

TL;DR

This paper investigates the saturation number, the size of the smallest maximal matching, in the corona product of graphs and certain chemistry-related graph constructions.

Contribution

It provides new insights into the saturation number for the corona product and specific chemistry-inspired graphs, expanding understanding of graph matchings.

Findings

Determined the saturation number for the corona product of specific graphs.

Analyzed the saturation number for graphs relevant in chemistry.

Provided formulas and bounds for the saturation number in these graph classes.

Abstract

Let $G=(V,E)$ be a simple connected graph. A matching $M$ in a graph $G$ is a collection of edges of $G$ such that no two edges from $M$ share a vertex. A matching $M$ is maximal if it cannot be extended to a larger matching in $G$. The cardinality of any smallest maximal matching in $G$ is the saturation number of $G$ and is denoted by $s(G)$. In this paper we study the saturation number of the corona product of two specific graphs. We also consider some graphs with certain constructions that are of importance in chemistry and study their saturation number.