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Connectivity and tree structure in finite graphs. (English) Zbl 1324.05104
Considering systems of separations in a graph $$G$$ that separate every pair of a given set of vertex sets that are themselves not separated by these separations, the authors determine conditions under which such a separation system contains a nested subsystem that still separates those sets and is invariant under automorphisms of $$G$$. As an application, it is shown that the $$k$$-blocks of a graph $$G$$ live in distinct parts of a tree-decomposition of $$G$$ of adhesion at most $$k$$, whose decomposition tree is invariant under the automorphisms of $$G$$. Under mild additional assumptions, these decompositions can be combined into one overall tree-decomposition that distinguishes all the $$k$$-blocks of a finite graph.

##### MSC:
 05C40 Connectivity 05C05 Trees 05C83 Graph minors
##### Keywords:
graph connectivity; graph decomposition; tree decomposition
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##### References:
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