On Hilbert bases of cuts

Luis Goddyn, Tony Huynh, Tanmay Deshpande

Research output: Contribution to journalArticleResearchpeer-review

2 Citations (Scopus)

Abstract

A Hilbert basis is a set of vectors X ⊆ ℝd such that the integer cone (semigroup) generated by X is the intersection of the lattice generated by X with the cone generated by X. Let ℋ be the class of graphs whose set of cuts is a Hilbert basis in RE (regarded as {0,1}-characteristic vectors indexed by edges). We show that ℋ is not closed under edge deletions, subdivisions, nor 2-sums. Furthermore, no graph having K6 \ e as a minor belongs to ℋ. This corrects an error in Laurent (1996). For positive results, we give conditions under which the 2-sum of two graphs produces a member of ℋ. Using these conditions we show that all K5-minor-free graphs are in ℋ, where K5 is the unique 3-connected graph obtained by uncontracting an edge of K5. We also establish a relationship between edge deletion and subdivision. Namely, if G′ is obtained from G ∈ ℋ by subdividing e two or more times, then G \ e ∈ ℋ if and only if G′ ∈ ℋ.

Original languageEnglish
Pages (from-to)721-728
Number of pages8
JournalDiscrete Mathematics
Volume339
Issue number2
DOIs
Publication statusPublished - 6 Feb 2016
Externally publishedYes

Keywords

  • Combinatorial optimization
  • Graph cut
  • Hilbert basis

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