# Topological and geometric graphs (NDMI095), LS 2021/2022

Jan Kyncl, Pavel Valtr, KAM (contact: surname - at - kam.mff.cuni.cz)
Tuesday at 14:00 in S10

If there is at least one student who does not understand Czech, the lectures will be in English; but you may ask questions in Czech during the lectures.

Topics planned:

• Introduction: drawings of graphs, topological graphs, geometric graphs
• The Hanani–Tutte theorem (weak, strong, and unified) and an algebraic algorithm for planarity testing
• The Jordan curve theorem and the nonplanarity of K_{3,3} (Thomassen's proof)
• Thrackles, thrackle conjecture
• Extremal results for topological and geometric graphs without forbidden substructures (k disjoint edges, k crossing edges)
• Structural properties of complete topological graphs (a Ramsey-type theorem)

Literature: (will be updated during the semester)

## Topics covered:

#### 22.2. (JK)

• Basic notions: topological graph, simple topological graph, geometric graph, plane graph, planar graph
• Statements of the Jordan curve theorem and the Jordan–Schönflies theorem
• Characterization of planar graphs: statements of Kuratovski's theorem, Wagner's theorem, Euler's formula, upper bound on the number of edges, Fáry's theorem
• Motivating question: what is the maximum possible number of edges in a simple topological graph (or a geometric graph) with no pair of disjoint edges?
• A "very weak" Hanani–Tutte theorem with a sketch of a proof

#### 1.3. (JK)

• The (strong) Hanani–Tutte theorem with a sketch of a proof using the Kuratowski theorem
• Algebraic algorithm for planarity testing, using the strong Hanani–Tutte theorem

#### 8.3. (JK)

• The weak Hanani–Tutte theorem, two proofs by redrawing
• Unified Hanani–Tutte theorem, first part of a proof (induction step for graphs with vertex connectivity 0, or 1 for the strong variant)

#### 15.3. (JK)

• Unified Hanani–Tutte theorem: proof for 3-connected graphs, sketch of a proof of the induction step for graphs with vertex connectivity 1 or 2

#### 22.3. (JK)

• Basic topological definitions: path-connected set, open set, closed set, interior, closure, boundary; simple polygonal arc, simple polygonal closed curve
• The Jordan curve theorem, Thomassen's proof, part 1: planar graphs have polygonal plane drawings, polygonal Jordan curve theorem, extension to polygonal drawings of K_2,3, nonplanarity of K_3,3 [Lemma 2.1 - Lemma 2.5 in the paper]

#### 29.3. (JK)

• The complement of a simple closed curve in the plane has at least two regions. [Proposition 2.6]
• The Jordan curve theorem, Thomassen's proof, part 2, including polygonal Euler's formula for 2-connected graphs and the Jordan arc theorem: the complement of a simple arc in the plane is connected. [Lemma 2.7 - Proposition 2.11]

#### 5.4. (JK)

• The complement of a simple closed curve in the plane has at most two regions. [Theorem 2.12]