Closed separator sets (Q2368589)

The paper is a study of \(\mathcal T(G)\), the set of smallest separators, from a new point of view. Let \(G\) be a simple graph \(G\) and \(\mathcal S \subseteq \mathcal T(G) \). For \(T\in \mathcal S\) the union of the vertex sets of at least one but not all components of \(G-T\) is called an \(\mathcal S\)-fragment. The paper defines and analyzes a closure operator on the separator sets. This notion leads to new proofs of several existing theorems and to new results as well, like: \(\bullet\) For every end \(B\) of a 2-critically connected graph there exists four disjoint fragments whose neighborhoods intersect \(B\). \(\bullet\) Every contraction critically \(k\)-connected graph admits two disjoint fragments \(A,B\) such that \(|A|+|B|\leq 2\lfloor k/4 \rfloor\). \(\bullet\) Let \(G\) be a contraction critically \(k\)-connected graph without fragments of cardinality less than \(k/4\). Then every atom is adjacent to at least two other atoms and \(G\) has at most 6 disjoint atoms.