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use decomposition into bi/tri-connected components and atoms for iterating over minimal separators #39383

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In #39341, we added an iterator over the minimal separators of a graph. Here we use decompositions into bi and tri connected components:

  • We first decompose the graph into biconnected components. Cut vertices are minimal separators. We then search for the minimal separators of each biconnected component
  • We then decompose each biconnected component into triconnected components. The "P" blocks of the SPQR-tree are minimal separators of order 2. The "S" blocks are cycle and each edge of its complement is a minimal separator. It remains to search for minimal separators in each "R" block (a triconnected component) using the main loop of the algorithm.

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@dcoudert
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We can also use a decomposition of the graph by clique minimal separators. It then remains to consider the atoms of the graph.

@dcoudert dcoudert changed the title use decomposition into bi/tri-connected components for iterating over minimal separators use decomposition into bi/tri-connected components and atoms for iterating over minimal separators Feb 10, 2025
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