Graphs with many independent vertex cuts
Yanan Hu, Xingzhi Zhan, Leilei Zhang

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 -connected graphs in which any two nonadjacent vertices form a vertex cut. We generalize this fact by proving that for every integer there exists a unique graph satisfying the following conditions: (1) is -connected; (2) the independence number of is greater than (3) any independent set of cardinality is a vertex cut of The edge version of this result does not hold. We also consider the problem when replacing independent sets by the periphery.
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Graph Labeling and Dimension Problems
