Trees of tangles in infinite separation systems

Abstract

We present infinite analogues of our splinter theorem from [15]. From these we derive several tree-of-tangles-type theorems for infinite graphs and infinite abstract separation systems.

Edge-connectivity and tree-structure in finite and infinite graphs

“…Key tool. The proof of our main result relies on a result from [20]. To state it, we shall need the following definitions.…”

“…When we wish to prove Theorem 1 without its additional properties, then it suffices to find a subset N ⊆ A that meets each A i and that is nested. One of the main results of [20] states that we can find N if the setup of the sets A i and their order function | • | satisfies a number of properties. The result can be applied even when I is infinite, and moreover it ensures that N is 'canonical' for the given setup.…”

“…Example 4.4. This example is a variation of [20,Example 4.9]. Consider the locally finite graph displayed in Figure 2.…”