Toughness of Recursively Partitionable Graphs

Authors

Calum Buchanan, University of VermontFollow
Brandon Du Preez, University of Cape TownFollow
K. E. Perry, Soka University of AmericaFollow
Puck Rombach, University of VermontFollow

Publication Date

December 2023

Abstract

A simple graph G = (V,E) on n vertices is said to be recursively partitionable (RP) if G ≃ K1, or if G is connected and satisfies the following recursive property: for every integer partition a₁, a₂, . . . , a_k of n, there is a partition {A₁,A₂, . . . ,A_k} of V such that each |A_i| = a_i, and each induced subgraph G[A_i] is RP (1 ≤ i ≤ k). We show that if S is a vertex cut of an RP graph G with |S| ≥ 2, then G−S has at most 3|S| − 1 components. Moreover, this bound is sharp for |S| = 3. We present two methods for constructing new RP graphs from old. We use these methods to show that for all positive integers s, there exist infinitely many RP graphs with an s-vertex cut whose removal leaves 2s + 1 components. Additionally, we prove a simple necessary condition for a graph to have an RP spanning tree, and we characterise a class of minimal 2-connected RP graphs.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

Recommended Citation

Buchanan, Calum; Du Preez, Brandon; Perry, K. E.; and Rombach, Puck (2023) "Toughness of Recursively Partitionable Graphs," Theory and Applications of Graphs: Vol. 10: Iss. 2, Article 4.
DOI: 10.20429/tag.2023.10204
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol10/iss2/4