Sparse graphs with an independent or foresty minimum vertex cut

TL;DR

This paper investigates the existence of independent or foresty minimum vertex cuts in connected graphs, establishing tight bounds on graph size relative to order for such cuts to exist.

Contribution

It proves tight bounds for the presence of independent and foresty minimum vertex cuts in connected graphs based on their size and order.

Findings

Graphs with size ≤ ⌊3n/2⌋ have independent minimum vertex cuts.

Graphs with size ≤ 2n have foresty minimum vertex cuts.

Results are optimal and extend previous characterizations.

Abstract

A connected graph is called fragile if it contains an independent vertex cut. In 2002 Chen and Yu proved that every connected graph of order $n$ and size at most $2n-4$ is fragile, and in 2013 Le and Pfender characterized the non-fragile graphs of order $n$ and size $2n-3.$ It is natural to consider minimum vertex cuts. We prove two results. (1) Every connected graph of order $n$ with $n\ge 7$ and size at most $\lfloor 3n/2\rfloor$ has an independent minimum vertex cut; (2) every connected graph of order $n$ with $n\ge 7$ and size at most $2n$ has a foresty minimum vertex cut. Both results are best possible.