remark & fix ref

main.tex

@@ -15,7 +15,7 @@
 % \maketitle
 
 \section{Ideal base packing}
-Try to generalize Thorup's ideal tree packing \cite{Thorup2008} to matroids. 
+Try to generalize Thorup's ideal tree packing \cite{Thorup_2008} to matroids. 
 Certainly it won't work on all matroids.
 The goal is to figure out some sufficient conditions and their relations with basepacking($\lambda\leq c \sigma$) and random contraction($\lambda \leq c \frac{|E|}{r(E)}$).
 
@@ -123,6 +123,9 @@ Let $X\subset E\setminus F^*$ be a independent set with rank $r(E)-r(F^*)$. Then
 \end{conjecture}
 \begin{remark}
 The intuition is that rigidity of $F^*\cup X$ only depends on the 1-thin cover of $F^*$ but not the base $B_{F^*}$.
+Consider a non-proper 1-thin cover where the rigid components come from those of 1-thin cover of $F^*$ and singleton elements of $X$. A proper 1-thin cover can be computed through coarsening.
+For a subset of rigid components $\mathcal P$, let $t=|\bigcup_{P\in \mathcal P} V[P]|$ be the number of vertices. If the number of edges $|\bigcup_{P\in \mathcal P} P|$ is at least $2t-3$ then we merge these components into a new one.
+One can see that in this process we do not care the actual base $B_{F^*}$ and only the 1-thin cover matters.
 \end{remark}
 
 \begin{comment}     % principal sequence of partition