The Menger-like property of the three-width of infinite graphs
The tree-width of a (possibly infinite) graph G is the minimum n such that G may be decomposed into a ``tree-structure'' of pieces each with at most \(n+1\) vertices. We prove that if G has tree-width n, then G can be decomposed in such a way that for every \(k\geq 0\) and any two pieces P and Q of this ``tree-structure'' either there are k disjoint paths between P and Q, or there is third piece R between P and Q with at most k-1 vertices. Our proof makes use of a finite version of this result proved by the second author and a compactness argument.
scientific article; zbMATH DE number 4105001
scientific article