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

ref.bib

@@ -32,17 +32,20 @@
 	pages = {1334--1353},
 }
 
-@article{Thorup2008,
+@inproceedings{Thorup_2008,
+	address = {Victoria British Columbia Canada},
 	title = {Minimum k-way cuts via deterministic greedy tree packing},
-	issn = {07378017},
-	url = {http://dl.acm.org/citation.cfm?id=1374402},
-	doi = {10.1145/1374376.1374402},
-	journal = {Proceedings of the 40th annual ACM symposium on …},
+	ISBN = {9781605580470},
+	url = {https://dl.acm.org/doi/10.1145/1374376.1374402},
+	DOI = {10.1145/1374376.1374402},
+	booktitle = {Proceedings of the fortieth annual ACM symposium on Theory of
+	             computing},
+	publisher = {ACM},
 	author = {Thorup, Mikkel},
 	year = {2008},
-	note = {ISBN: 9781605580470},
-	keywords = {k-way cuts, tree packing},
-	pages = {159--165},
+	month = may,
+	pages = {159–166},
+	language = {en},
 }
 
 @article{thorup_fully-dynamic_2007,
@@ -59,5 +62,7 @@
 	month = feb,
 	year = {2007},
 	pages = {91--127},
-	file = {Thorup_2007_Fully-Dynamic Min-Cut.pdf:/Users/congyu/Zotero/storage/Q329VHH3/Thorup_2007_Fully-Dynamic Min-Cut.pdf:application/pdf},
+	file = {Thorup_2007_Fully-Dynamic
+	        Min-Cut.pdf:/Users/congyu/Zotero/storage/Q329VHH3/Thorup_2007_Fully-Dynamic
+	        Min-Cut.pdf:application/pdf},
 }