A topological approach to evasiveness

A graph property G is a collection of graphs closed under isomorphism. G is said to be evasive if, for every possible local search strategy, there is at least one graph for which membership in G cannot be decided until the entire graph has been searched. Most graph properties that one can think of, but not all, turn out to be evasive. A conjecture of Aanderaa, Karp and Rosenberg asserts that every monotone (closed under adding edges) property is evasive (except for two trivial exceptions). I’ll explain how topological ideas introduced by Kahn, Saks and Sturtevant, in particular the study of fixed points of simplicial maps, have helped to make significant progress towards the AKR conjecture. These notes are based in part on the original paper of Kahn, Saks and Sturtevant [2] and in part on lecture notes of Lovász and Young [4].