Symmetry in discrete structures: graphs, groups and algorithms

• Autores: Pascal Schweitzer
• Localización: Discrete Mathematics Days 2022 / Luis Felipe Tabera Alonso (ed. lit.), 2022, ISBN 978-84-19024-02-2, págs. 8-8
• Idioma: inglés
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• Resumen

  • The concept of symmetry is ubiquitous in the study of discrete combinatorial objects. The ensemble ofsymmetries of an object G is its automorphism group Aut(G). It provides us with structural informationof the object. The group also captures how its symmetries relate to one another. While symmetries areinteresting to study in their own right, symmetries also have direct applications in diverse areas thatmake use of combinatorial objects.In the talk, I will describe various facets of research surrounding the structure, detection andapplication of symmetries.• Regarding the structure of symmetry, I will discuss automorphism groups of graphs in minor-closedgraph classes. Examples of such graph classes include the class of trees, planar graphs, graphsof bounded genus (i.e., graphs embeddable into a fixed surface without crossings), and graphsof bounded tree width. While a classic result of Frucht shows that every finite group is theautomorphism group of a finite graph, Babai showed that with a fixed graph class that excludesa fixed minor not all groups can be represented as automorphism groups. The talk will discussgeneral structure theorems for the automorphism group of minor-closed graph classes.• Regarding detection of symmetry, it is well known that the so-called graph isomorphism problem,which asks us to decide whether two given graphs are isomorphic, is universal in the following sense.The computation of the automorphism group of an arbitrary (explicitly given) combinatorialobject reduces, in polynomial time, to the graph isomorphism problem. The isomorphism problemin turn reduces to a related problem, the problem of computing a canonical labeling. In somecontexts this is referred to as computing a normal form of a given graph. The talk will discuss somerecent advances in the research on the theory and implementation of practical graph isomorphismsolvers.• Regarding application of symmetry, I will focus on a particular type of application in finite modeltheory. Specifically, I will touch on the quest for a logic capturing Ptime and discuss how itrelates to the graph isomorphism and graph canonization problems.This talk is in part an overview over various results obtained within the ERC-Project EngageS(https://www.mathematik.tu-darmstadt.de/EngageS) and based on joint work with numerousco-authors.