Course 2-38-1, year 2024-2025.

Teachers: Luca Castelli Aleardi (École Polytechnique) and Éric Colin de Verdière (CNRS & Université Gustave Eiffel).

Other teachers not teaching this year: Vincent Cohen-Addad (Google Zürich), Arnaud de Mesmay (CNRS & Université Gustave Eiffel), and Vincent Pilaud (CNRS & École Polytechnique, responsible teacher for the course).

Final exam on Dec. 4, 8:45-11:45am, usual room (1004 of Sophie Germain building). The exam will be in two parts, one for each teacher. Please prepare two sets of answer sheets, one for each part. Allowed documents: the course notes and handwritten documents. No electronic devices. The exam will have a French and an English version; answers can be given in either language.

Second homework due Nov. 13 (optional) here.

First homework due Oct. 16 (optional) here.

When? First period, Wednesdays, from 8:45 to 11:45, starting September 18. No lecture on October 30.

Where? Sophie Germain room 1004.

Language. Lectures will be given in French by default, or in English upon request of at least three persons who do not understand French, and if nobody objects. Questions can be raised in English or in French. Lecture notes will be available in English. The exam may be provided both in English and in French if requested.

Evaluation. The evaluation is done with the final exam. Two exercises sheets will be proposed during the course period; if solved and given to the teacher (either a sheet written with a pen, or a LaTeX-formatted electronic version), they will give some extra credit for the final grade.

Prerequisites. None.

Algorithms and combinatorics for graphs are a major theme in computer science. In this course, we study various aspects of this theme in the case of graphs arising in geometric settings. Examples include planar graphs (of course), graphs drawn without crossings on topological surfaces, and graphs of polytopes and other combinatorial structures. The course is therefore at the frontier of graph algorithms, combinatorics, and computational geometry.

Following this course is a good opportunity

• to see some relatively standard tools in algorithms and combinatorics applied in geometric settings,
• leading to results at the edge of current research, and
• to learn about some fundamental objects such as polytopes and surfaces, which appear in various contexts (optimization, discrete mathematics, topological graph theory).

• Graphs drawn in the plane:
  • basics: combinatorial representations of planar graphs, topology, duality, Euler's formula;
  • Schnyder woods;
  • planarity testing and Tutte embedding;
  • efficient algorithms for planar graphs;
• Graphs on surfaces:
  • classification theorem for surfaces up to homeomorphism;
  • topological algorithms for graphs on surfaces: shortest loops and systems of loops
  • Homotopy testing, and perhaps a few more things if time allows.

LCA's lectures: Lecture 1, Lecture 4, Lecture 6, Lecture 8, course notes on Schnyder woods.

ÉCdV's lectures: course notes (only a subset will be treated)

• The course notes (since no book covers all these topics).
• Jesus A. De Loera, Jörg Rambau, and Francisco Santos. Triangulations: Structures for Algorithms and Applications, volume 25 of Algorithms and Computation in Mathematics. Springer Verlag, 2010.
• Stefan Felsner. Geometric graphs and arrangements. Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Wiesbaden, 2004. Some chapters from combinatorial geometry.
• Bojan Mohar and Carsten Thomassen. Graphs on surfaces. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, 2001.
• Günter M. Ziegler. Lectures on polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995.
• Related courses have been taught elsewhere, with materials available online. See, e.g., Jeff Erickson and Francis Lazarus and Arnaud de Mesmay

The course has some connections with the following ones:

• 2-10 Algorithmic aspects of combinatorics;
• 2-14-1 Computational geometry learning;
• 2-29-1 Graph algorithms;
• 2-39 Computer Graphics and Scientific Visualization.