The Structure of Minimum Vertex Cuts

TL;DR

This paper studies the structure of minimum vertex cuts in graphs, classifies their types, and introduces efficient data structures and algorithms for connectivity queries and cut computations.

Contribution

It provides a new classification of minimum vertex cuts and develops a simple, space-efficient data structure for quick connectivity queries in highly connected graphs.

Findings

Introduces an $O( abla ext{space})$ data structure for pairwise connectivity queries.

Shows how to compute the closest $ abla$-cut to each vertex in near linear time.

Classifies relationships between multiple minimum vertex cuts.

Abstract

In this paper we continue a long line of work on representing the cut structure of graphs. We classify the types minimum vertex cuts, and the possible relationships between multiple minimum vertex cuts. As a consequence of these investigations, we exhibit a simple $O(\kappa n)$-space data structure that can quickly answer pairwise $(\kappa+1)$-connectivity queries in a $\kappa$-connected graph. We also show how to compute the "closest" $\kappa$-cut to every vertex in near linear $\tilde{O}(m+poly(\kappa)n)$ time.