Closed Separator Sets

A smallest separator in a finite, simple, undirected graph G is a set SV (G) such that GS is disconnected and |S|=κ(G), where κ(G) denotes the connectivity of G.

A set S of smallest separators in G is defined to be closed if for every pair S,TS, every component C of GS, and every component S of GT intersecting C either X(C,D) := (V (C) ∩ T) ∪ (TS) ∪ (SV (D)) is in S or |X(C,D)| > κ(G). This leads, canonically, to a closure system on the (closed) set of all smallest separators of G.

A graph H with \( V{\left( H \right)} \subseteq V{\left( G \right)},\;E{\left( H \right)} \cap E{\left( G \right)} = \emptyset \) is defined to be S-augmenting if no member of S is a smallest separator in GH:=(V (G) ∪ V (H), E(G) ∪ E(H)). It is proved that if S is closed then every minimally S-augmenting graph is a forest, which generalizes a result of Jordán.

Several applications are included, among them a generalization of a Theorem of Mader on disjoint fragments in critically k-connected graphs, a Theorem of Su on highly critically k-connected graphs, and an affirmative answer to a conjecture of Su on disjoint fragments in contraction critically k-connected graphs of maximal minimum degree.

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Correspondence to Matthias Kriesell.

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Kriesell, M. Closed Separator Sets. Combinatorica 25, 575–598 (2005). https://doi.org/10.1007/s00493-005-0035-1

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Mathematics Subject Classification (2000):

  • 05C40