On Hilbert bases of cuts

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• 作者：Luis Goddyn^a ; ^{goddyn@sfu.ca" class="auth_mail" title="E-mail the corresponding author} ; Tony Huynh^b ; ^{tony.bourbaki@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Tanmay Deshpande^a ; ^{tanmaydesh5886@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Hilbert basis ; Graph cut ; Combinatorial optimization
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 February 2016
• 年：2016
• 卷：339
• 期：2
• 页码：721-728
• 全文大小：419 K

文摘

A Hilbert basis   is a set of vectors md5=b435561ef964ffde07f77aca55f847bf" title="Click to view the MathML source">X⊆R^d$X\subseteq {\mathbb{R}}^d$ such that the integer cone (semigroup) generated by md5=89c3c6c07683e3160c54792ab6bb0354" title="Click to view the MathML source">X$X$ is the intersection of the lattice generated by md5=89c3c6c07683e3160c54792ab6bb0354" title="Click to view the MathML source">X$X$ with the cone generated by md5=89c3c6c07683e3160c54792ab6bb0354" title="Click to view the MathML source">X$X$. Let md5=5b00ec8123b6ba9bd4a722f0f3ffd47f" title="Click to view the MathML source">ℋ$\mathscr{H}$ be the class of graphs whose set of cuts is a Hilbert basis in md5=6f78ac7977a4fe2a470aae8cf269487d" title="Click to view the MathML source">R^E${\mathbb{R}}^E$ (regarded as md5=4854a3393cde6dcf748121e3af98e2f9" title="Click to view the MathML source">{0,1}$\{0,1\}$-characteristic vectors indexed by edges). We show that md5=5b00ec8123b6ba9bd4a722f0f3ffd47f" title="Click to view the MathML source">ℋ$\mathscr{H}$ is not closed under edge deletions, subdivisions, nor 2-sums. Furthermore, no graph having md5=848aa952678e3c3adfdccfc68021be5c" title="Click to view the MathML source">K₆∖e$K_6\setminus e$ as a minor belongs to md5=5b00ec8123b6ba9bd4a722f0f3ffd47f" title="Click to view the MathML source">ℋ$\mathscr{H}$. 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 md5=5b00ec8123b6ba9bd4a722f0f3ffd47f" title="Click to view the MathML source">ℋ$\mathscr{H}$. Using these conditions we show that all md5=69e62aeb8269593ea14ebf389519dae4">$K_5^{\&<font\; color="red">perp</font>;}$-minor-free graphs are in md5=5b00ec8123b6ba9bd4a722f0f3ffd47f" title="Click to view the MathML source">ℋ$\mathscr{H}$, where md5=69e62aeb8269593ea14ebf389519dae4">$K_5^{\&<font\; color="red">perp</font>;}$ is the unique 3-connected graph obtained by uncontracting an edge of md5=7270fdad0831188fb3fe0e250cae7317" title="Click to view the MathML source">K₅$K_5$. We also establish a relationship between edge deletion and subdivision. Namely, if md5=98a06e38af76bd8372c47f56a5f7e95b" title="Click to view the MathML source">G^′$G^{\prime }$ is obtained from md5=bb27505092668b7e03b4f8d6c5b329c2" title="Click to view the MathML source">G∈ℋ$G\in \mathscr{H}$ by subdividing md5=fb3b3841861cc2f54f758ebb5baafe09" title="Click to view the MathML source">e$e$ two or more times, then md5=23ce2ce00c83100ab90ba2bcc7a2e237" title="Click to view the MathML source">G∖e∈ℋ$G\setminus e\in \mathscr{H}$ if and only if md5=ecbe4445e1db56f135b1584758b43307" title="Click to view the MathML source">G^′∈ℋ$G^{\prime }\in \mathscr{H}$.