From a7cacdd04d8901a96bdc39824e9de5e8fb940bab Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Tue, 25 Nov 2025 12:15:35 +0800 Subject: [PATCH] z --- main.tex | 5 ++--- 1 file changed, 2 insertions(+), 3 deletions(-) diff --git a/main.tex b/main.tex index 79f425f..bb73d80 100644 --- a/main.tex +++ b/main.tex @@ -125,8 +125,7 @@ Let $X\subset E\setminus F^*$ be a independent set with rank $r(E)-r(F^*)$. Then The intuition is that rigidity of $F^*\cup X$ only depends on the 1-thin cover of $F^*$ but not the base $B_{F^*}$. \end{remark} -\newpage - +\begin{comment} % principal sequence of partition \section{Principal sequence of partition on graphs} For a graph $G=(V,E)$ with edge capacity $c:V\to \Z_+$, the strength $\sigma(G)$ is defined as $\sigma(G)=\min_{\Pi}\frac{c(\delta(\Pi))}{|\Pi|-1}$, where $\Pi$ is any partition of $V$, $|\Pi|$ is the number of parts in the partition and $\delta(\Pi)$ is the set of edges between parts. Note that an alternative formulation of strength (using graphic matroid rank function) is $\sigma(G)=\min_{F\subset E} \frac{c(E-F)}{r(E)-r(F)}$, which in general is the fractional optimum of matroid base packing. @@ -144,7 +143,7 @@ The number of breakpoints on $L(\lambda)$ is at most $n-1$. That is $c(E\setminus F')-\lambda(r(E)-r(F')+1)>-\lambda$ for all $F'\subsetneq E$. We are interested in the upperbound $\e$ of $\lambda$ such that the optimal $F$ is a proper subset of $E$ when $\lambda >\e$. Therefore, the upperbound is $\e=\min_{F\subsetneq E}\frac{c(E\setminus F)}{r(E)-r(F)}$, which is exactly the strength. \end{enumerate} - +\end{comment} \bibliographystyle{plain} \bibliography{ref}