# Random stuff, 2016-01-08

Doing some random readings, also trying to revive the blog :(

• Conjecture [Thomas].  For any t, any sufficiently large t-connected graph with no $K_t$-minor can be made planar by removing exactly t-5 vertices.
The case when t=6 has been proven. [Kawarabayashi-Norine-Thomas-Wollan ’12]
• In a subgraph-closed graph family, having polynomial expansion is equivalent to having sublinear separator.  [Dvořák-Norine ’15]
• Richter-Thomassen Conjecture, now Pach-Rubin-Tardos Theorem, states that the total number of intersections between n pairwise intersecting Jordan curves in the plane, no three pass through the same point, is at least $(1-o(1))n^2$.
Main Theorem [Pach-Rubin-Tardos ’15].  Consider the above set of Jordan curves.  Then the number of crossing points is $\Omega(\log\log n)^{1/8}$ times the number of touching points, those that are the only intersection between two Jordan curves.
• Further results on arc and bar k-visibility graphs by Sawhney and Weed, 2016.
• Minimum cut of directed planar graphs in O(n log log n) time by Mozes, Nikolaev, Nussbaum, and Weimann, 2015.
• The weakly simple polygon result has been extended to geometric intersection numbers. [Despré-Lazarus ’15]