Paths, Ends and The Separation Problem for Infinite Graphs

TL;DR

This paper introduces the Separation Problem for infinite graphs, proves its decidability under certain conditions, and explores its implications for graph properties like ends and Eulerian paths, providing complexity characterizations.

Contribution

It establishes the decidability of the Separation Problem for highly computable infinite graphs with finitely many ends and analyzes its impact on Eulerian Path complexity.

Findings

Separation Problem is decidable for certain infinite graphs.

Eulerian Path complexity depends on the number of ends.

Characterization of the problem's complexity in a uniform setting.

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 K\"onig'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.