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Paths, Ends and The Separation Problem for Infinite Graphs

Nicanor Carrasco-Vargas, Valentino Delle Rose, Cristóbal Rojas

TL;DR

The paper introduces and analyzes the Separation Problem for infinite graphs from a computability perspective, proving decidability for uniformly highly computable graphs with finitely many ends and connecting this to effective versions of König's Lemma. It develops a uniform framework with Sep($G$) and Sepmax($G$), clarifies their relation to Ends($G$), and uses these tools to study infinite-path computability and the Eulerian Path Problem, showing that end-counting is a central source of complexity. It provides precise complexity classifications for fixed versus input graphs, and demonstrates uniform reductions between separation, ends counting, and path-extensions, including tree-specific equivalences. Overall, the work advances effective graph theory by isolating ends as a key determinant of algorithmic behavior for fundamental properties on infinite graphs.

Abstract

We introduce and study the Separation Problem for infinite graphs, which involves determining whether a connected graph splits into at least two infinite connected components after the removal of a given finite set of edges. We prove that this problem is decidable for every highly computable graph with finitely many ends. Using this result, we demonstrate that König's Infinity Lemma is effective for such graphs. We also apply it to analyze the complexity of the Eulerian Path Problem for infinite graphs, showing that much of its complexity arises from counting ends. Indeed, the Eulerian Path Problem becomes strictly easier when restricted to graphs with a fixed number of ends. Under this restriction, we provide a complete characterization of the problem. Finally, we study the Separation Problem in a uniform setting (i.e., where the graph is also part of the input) and offer a nearly complete characterization of its complexity and its relationship to counting the number of ends.

Paths, Ends and The Separation Problem for Infinite Graphs

TL;DR

The paper introduces and analyzes the Separation Problem for infinite graphs from a computability perspective, proving decidability for uniformly highly computable graphs with finitely many ends and connecting this to effective versions of König's Lemma. It develops a uniform framework with Sep() and Sepmax(), clarifies their relation to Ends(), and uses these tools to study infinite-path computability and the Eulerian Path Problem, showing that end-counting is a central source of complexity. It provides precise complexity classifications for fixed versus input graphs, and demonstrates uniform reductions between separation, ends counting, and path-extensions, including tree-specific equivalences. Overall, the work advances effective graph theory by isolating ends as a key determinant of algorithmic behavior for fundamental properties on infinite graphs.

Abstract

We introduce and study the Separation Problem for infinite graphs, which involves determining whether a connected graph splits into at least two infinite connected components after the removal of a given finite set of edges. We prove that this problem is decidable for every highly computable graph with finitely many ends. Using this result, we demonstrate that König's Infinity Lemma is effective for such graphs. We also apply it to analyze the complexity of the Eulerian Path Problem for infinite graphs, showing that much of its complexity arises from counting ends. Indeed, the Eulerian Path Problem becomes strictly easier when restricted to graphs with a fixed number of ends. Under this restriction, we provide a complete characterization of the problem. Finally, we study the Separation Problem in a uniform setting (i.e., where the graph is also part of the input) and offer a nearly complete characterization of its complexity and its relationship to counting the number of ends.
Paper Structure (9 sections, 24 theorems, 6 equations, 1 table)

This paper contains 9 sections, 24 theorems, 6 equations, 1 table.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Preliminaries
  4. The Separation problem
  5. The complexity of $\mathbf{Sep}({G})$ and $\mathbf{Sepmax}({G})$ for a fixed graph $G$
  6. The complexity of $\mathbf{Sep}({G})$ and $\mathbf{Sepmax}({G})$ when $G$ is in the input
  7. The relationship between $\mathbf{Sep}({G})$, $\mathbf{Sepmax}({G})$ and $\mathbf{Ends}({G})$
  8. Computing infinite paths by separation
  9. On the complexity of the Eulerian path problem

Key Result

Theorem 2.1

A set $A \subseteq \mathbb{N}$ is $\Delta_2^0$ if and only if $A$ has a computable approximation, namely a computable function $f \colon \mathbb{N}^2 \to \{0, 1 \}$ such that, for all $n$:

Theorems & Definitions (53)

  • Theorem 2.1: Shoenfield's Limit Lemma limit-lemma
  • Definition 2.2
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • ...and 43 more