# T-joins in infinite graphs as edge-disjoint system of paths matching the vertices in $ T $

## TL;DR

This paper characterizes infinite connected graphs that admit a T-join for any infinite subset T of vertices, extending finite graph results to the infinite case and analyzing the conditions under which T-joins exist or can be modified.

## Contribution

It provides a characterization of infinite graphs with T-joins for any infinite subset T, extending classical finite graph results to the infinite setting.

## Key findings

- Characterization of infinite graphs with T-joins for any infinite T
- Extension of finite T-join properties to infinite graphs
- Invariance of T-join existence under vertex removal or addition in connected graphs

## Abstract

We characterize the class of infinite connected graphs $ G $ for which there exists a $ T $-join for any choice of an infinite $ T \subseteq V(G) $. We also show that the following well-known fact remains true in the infinite case. If $ G $ is connected and does not contain a $ T $-join, then it will if we either remove an arbitrary vertex from $ T $ or add any new vertex to $ T $.