From a0d18b0bfe4fe199a1e27deaa042774a5af8593b Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Tue, 16 Dec 2025 15:36:55 +0800 Subject: [PATCH] z --- main.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/main.tex b/main.tex index deef323..6ce3df9 100644 --- a/main.tex +++ b/main.tex @@ -216,7 +216,7 @@ One can see that in this process we do not care the actual base $B_{F^*}$ and on \autoref{conj:dist} is true when $M$ is a 2D rigidity matroid. \end{conjecture} -Try to minic Thorup's proof for graphic matroids. As \autoref{lem:partition} states, spans form a partition on $E-F$. +Try to minic Thorup's proof for graphic matroids. It follows from \autoref{lem:partition} that spans form a partition on $E-F$. Note that for graphic matroids the number of spans is $O((r(E)-r(F))^2)$. For any spanning tree $T$, the number of spans hitting $T$ is exactly $r(E)-r(F)$ and these spans have a nice structure. If we contract each component in $G[F]$ to a vertex and consider spans a set of parallel edges, then the set of spans hitting $T$ is a tree (with parallel edges) in $G[F]$. For rigidity matroids, the number of rigid components in $F$ cannot be bounded by $r(E)-r(F)$.\footnote{Since the 1-thin cover inducing a cocircuit can have any number of rigid components.}