Today’s topic: The Planar Separator Theorem.

What does it say? A planar graph on n vertices has a small “separator,” meaning we can partition the set of vertices into three sets A, B, and S such that:

1) |A|, |B| <= 2n/3

2) |S| <= 4 sqrt{n}

3) There is no edge between A and B.

We started the proof of the theorem today and solved the first few cases. If the graph is already disconnected, and the components are small enough, there isn't much to do.

If one of the components is too big, or if the graph itself is connected, we actually have to find a separator set. The process of finding such a separator starts with a BFS traversal and labeling of the levels.

Please review the boards and the book before Wednesday's class. We'll finish up the theorem next time so bring all of your questions (or ask them here)!

We also have a small correction to the homework assignment which I will correct in its own post.

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