Abstract
This paper proposes a new proof of Dilworth's theorem. The proof is based upon the minflow/maxcut property in flow networks. In relation to this proof, a new method to find both a Dilworth decomposition and a maximal antichain is presented.
Highlights
Several proofs are known for Dilworth’s theorem. is theorem says that, in a poset PP, a maximal antichain and a minimal path cover have equal size
In the current paper a shortcut is proposed between max ow/mincut and an optimal path cover jointly with an antichain
An antichain in a directed acyclic graph DD(VVV VV) is a set of nodes, no two of which are included in any path of DD(VVV VV)
Summary
Several proofs are known for Dilworth’s theorem. is theorem says that, in a poset PP, a maximal antichain and a minimal path cover have equal size. Is theorem says that, in a poset PP, a maximal antichain and a minimal path cover have equal size. A er Dilworth’s seminal paper [1] a “Note” [2] was published containing an algorithmic proof, that is, a proof which gives a method to nd a combination of a maximal antichain and a minimal path cover. Dilworth’s theorem is proved in [2] using König’s theorem stating that, in a bipartite graph, a maximal matching and a minimal vertex cover have equal size. An antichain in a directed acyclic graph DD(VVV VV) is a set of nodes, no two of which are included in any path of DD(VVV VV). In a poset the size of a maximal antichain equals the size of a minimal path cover.
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