Path-closed sets

Abstract

Given a digraphG = (V, E), call a node setTV path-closed ifv, v′ εT andw εV is on a path fromv tov′ impliesw εT. IfG is the comparability graph of a posetP, the path-closed sets ofG are the convex sets ofP. We characterize the convex hull of (the incidence vectors of) all path-closed sets ofG and its antiblocking polyhedron inR v, using lattice polyhedra, and give a minmax theorem on partitioning a given subset ofV into path-closed sets. We then derive good algorithms for the linear programs associated to the convex hull, solving the problem of finding a path-closed set of maximum weight sum, and prove another min-max result closely resembling Dilworth’s theorem.

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Gröflin, H. Path-closed sets. Combinatorica 4, 281–290 (1984). https://doi.org/10.1007/BF02579138

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AMS subject classification (1980)

  • 05 C 20
  • 05 C 38
  • 06 A 10
  • 52 A 25
  • 90 B 10
  • 90 C 05