Saturation Numbers for Minors

TL;DR

This paper investigates the saturation numbers for graph minors, providing bounds and demonstrating that certain generalized Petersen graphs are minor saturated for specific complete graph minors.

Contribution

It offers new bounds on saturation numbers for minors and identifies specific graphs that are minor saturated for certain complete minors.

Findings

Established bounds for saturation numbers for minors.

Identified generalized Petersen graphs as minor saturated for specific minors.

Contributed to understanding the structure of minor-saturated graphs.

Abstract

The saturation number $\text{sat}(n,\mathcal{F})$ is the minimum number of edges in any graph which does not contain a member of $\mathcal{F}$ as a subgraph, but will if any edge is added. We give a few upper and lower bounds for saturation numbers for minors. In particular, we shall show that certain Generalized Petersen Graphs are $K^r$-minor saturated for $6\le r\le 8$.