Graphs with many independent vertex cuts

TL;DR

This paper characterizes unique k-connected graphs with large independence number where all size-k independent sets are vertex cuts, extending known properties of cycles to higher connectivity levels.

Contribution

It generalizes a known characterization of cycles to all k-connected graphs with large independence number, introducing a new class of graphs with this property.

Findings

Existence of a unique k-connected graph with specified properties for each k≥3

The edge version of the property does not hold

Consideration of the problem with periphery instead of independent sets

Abstract

The cycles are the only $2$-connected graphs in which any two nonadjacent vertices form a vertex cut. We generalize this fact by proving that for every integer $k\ge 3$ there exists a unique graph $G$ satisfying the following conditions: (1) $G$ is $k$-connected; (2) the independence number of $G$ is greater than $k;$ (3) any independent set of cardinality $k$ is a vertex cut of $G.$ The edge version of this result does not hold. We also consider the problem when replacing independent sets by the periphery.