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Splitting-off in Hypergraphs

Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, Shubhang Kulkarni

TL;DR

This work extends the classical splitting-off operation from graphs to hypergraphs by defining a local-connectivity-preserving complete $h$-splitting-off at a vertex and proving its existence, along with a strongly polynomial-time algorithm for weighted hypergraphs. It then leverages this operation to obtain two key results: a constructive characterization of $k$-hyperedge-connected hypergraphs via a pair of elementary operations, and an alternative, hypergraph-based proof of an approximate min–max relation for Steiner rooted-connected orientations in graphs and hypergraphs. The methods blend hypergraph submodularity, Bernáth–Király type function-cover techniques, and lifting arguments to connect hypergraph properties back to graph orientations and Menger-type results. The findings advance both theory and algorithm design for hypergraph connectivity and orientation problems, with potential implications for hypergraph optimization and network design.

Abstract

The splitting-off operation in undirected graphs is a fundamental reduction operation that detaches all edges incident to a given vertex and adds new edges between the neighbors of that vertex while preserving their degrees. Lovász (1974) and Mader (1978) showed the existence of this operation while preserving global and local connectivities respectively in graphs under certain conditions. These results have far-reaching applications in graph algorithms literature. In this work, we introduce a splitting-off operation in hypergraphs. We show that there exists a local connectivity preserving complete splitting-off in hypergraphs and give a strongly polynomial-time algorithm to compute it in weighted hypergraphs. We illustrate the usefulness of our splitting-off operation in hypergraphs by showing two applications: (1) we give a constructive characterization of $k$-hyperedge-connected hypergraphs and (2) we give an alternate proof of an approximate min-max relation for max Steiner rooted-connected orientation of graphs and hypergraphs (due to Király and Lau (Journal of Combinatorial Theory, 2008; FOCS 2006)). Our proof of the approximate min-max relation for graphs circumvents the Nash-Williams' strong orientation theorem and uses tools developed for hypergraphs.

Splitting-off in Hypergraphs

TL;DR

This work extends the classical splitting-off operation from graphs to hypergraphs by defining a local-connectivity-preserving complete -splitting-off at a vertex and proving its existence, along with a strongly polynomial-time algorithm for weighted hypergraphs. It then leverages this operation to obtain two key results: a constructive characterization of -hyperedge-connected hypergraphs via a pair of elementary operations, and an alternative, hypergraph-based proof of an approximate min–max relation for Steiner rooted-connected orientations in graphs and hypergraphs. The methods blend hypergraph submodularity, Bernáth–Király type function-cover techniques, and lifting arguments to connect hypergraph properties back to graph orientations and Menger-type results. The findings advance both theory and algorithm design for hypergraph connectivity and orientation problems, with potential implications for hypergraph optimization and network design.

Abstract

The splitting-off operation in undirected graphs is a fundamental reduction operation that detaches all edges incident to a given vertex and adds new edges between the neighbors of that vertex while preserving their degrees. Lovász (1974) and Mader (1978) showed the existence of this operation while preserving global and local connectivities respectively in graphs under certain conditions. These results have far-reaching applications in graph algorithms literature. In this work, we introduce a splitting-off operation in hypergraphs. We show that there exists a local connectivity preserving complete splitting-off in hypergraphs and give a strongly polynomial-time algorithm to compute it in weighted hypergraphs. We illustrate the usefulness of our splitting-off operation in hypergraphs by showing two applications: (1) we give a constructive characterization of -hyperedge-connected hypergraphs and (2) we give an alternate proof of an approximate min-max relation for max Steiner rooted-connected orientation of graphs and hypergraphs (due to Király and Lau (Journal of Combinatorial Theory, 2008; FOCS 2006)). Our proof of the approximate min-max relation for graphs circumvents the Nash-Williams' strong orientation theorem and uses tools developed for hypergraphs.
Paper Structure (37 sections, 39 theorems, 33 equations, 3 figures, 1 algorithm)

This paper contains 37 sections, 39 theorems, 33 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Hypergraphs.
  3. Notation.
  4. Hypergraph Splitting-off.
  5. Application 1: Constructive characterization of $k$-hyperedge-connected hypergraphs.
  6. Application 2.1: Steiner Rooted $k$-arc-connected Orientation of Graphs.
  7. Application 2.2: Steiner Rooted $k$-hyperarc-connected Orientation of Hypergraphs.
  8. Proof Technique for Theorem \ref{['thm:k-EC-hypergraph-characterization']}
  9. Proof Technique for Theorems \ref{['thm:rooted-steiner-orientation-in-graphs']} and \ref{['thm:rooted-steiner-orientation-in-hypergraphs']}
  10. Proof Technique for Theorem \ref{['thm:complete-splitting-off']}
  11. Proof Technique for \ref{['thm:WeakToStrongCover:main']}
  12. Preliminaries.
  13. Algorithm of Bernath-Kiraly.
  14. Preprocessing for Additional Structure.
  15. Our Algorithm.
  16. ...and 22 more sections

Key Result

Theorem 1.1

Given a hypergraph $(G=(V, E), w_G: E\rightarrow \Z_+)$ and a vertex $s\in V$, there exists a strongly polynomial-time algorithm to find a local connectivity preserving complete h-splitting-off at $s$ from $(G, w_G)$.

Figures (3)

  • Figure 1: Example of (local connectivity preserving) complete h-splitting-off at a vertex $s$ from a hypergraph. Consider the leftmost hypergraph where all hyperedge weights are one and the vertex $s$ is as labeled. Operations (I) and (II) correspond to h-merge almost-disjoint hyperedges operations and Operation (III) corresponds to an h-trim hyperedge operation.
  • Figure 2: An example showing that global connectivity preserving complete g-splitting-off at a vertex from a graph may not exist. All edge weights are one and the vertex $s$ is as labeled.
  • Figure 3: An example of a $(4,2)$-pinching operation. Here, $t_1 = t_2 = 2$.

Theorems & Definitions (107)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Remark 1.1
  • Definition 1.3
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3: Király and Lau Kiraly-Lau
  • Remark 1.3
  • Theorem 1.4: Király and Lau Kiraly-Lau
  • ...and 97 more