The smallest nonevasive graph property

A property of n-vertex graphs is called evasive if every algorithm testing this property by asking questions of the form “is there an edge between vertices u and v” requires, in the worst case, to ask about all pairs of vertices. Most “natural” graph properties are either evasive or conjectured to be such, and of the few examples of nontrivial nonevasive properties scattered in the literature the smallest one has n = 6. We exhibit a nontrivial, nonevasive property of 5-vertex graphs and show that it is essentially the unique such with n ≤ 5. Evasiveness is a complexity-theoretic concept defined via the following combinatorial game. Two players, Alice and Bob, first fix a number n and a property P of n-vertex graphs. Bob wants to find out if some unknown graph G, secretly chosen by Alice, has the property P, by asking Alice one by one if a particular pair of vertices forms an edge. Alice wins if she can force Bob to ask about all the ( n 2 ) pairs before he knows if G ∈ P. Bob wins if he can decide the membership of G in P after at most ( n 2 ) −1 questions. Of course there is no reason why Alice should fix any particular graph in advance — she can adapt her answers so as to force Bob to ask the maximal number of questions. We say P is evasive (or elusive) if Alice has a winning strategy; it is nonevasive if Bob does. For example, the simple property of “being the complete graph” is evasive. Alice’s strategy is to say “Yes” to Bob’s first ( n 2 ) − 1 questions, at which point he is still not sure if G is complete or not. To be more precise, for a fixed natural number n let Gn be the set of isomorphism classes of n-vertex simple, unlabeled graphs. A property of n-vertex graphs is just an arbitrary subset P ⊆ Gn. We usually say “a graph G has property P” (e.g. G is connected, G is a tree, G has a Hamiltonian cycle etc.) meaning “G is isomorphic to one of the graphs in P”. For every n there are two trivial nonevasive properties, P = ∅ and P = Gn, for which Bob wins without asking any questions at all. More generally, P is evasive if and only if so is Gn\P, with Bob playing the same strategy. Evasiveness is a classical notion which arose as a way of measuring the decisiontree complexity of boolean functions. The lecture notes [5] are an excellent introduction to this general topic. Here it suffices to say that most “natural” graph properties, for example connectedness, planarity, triangle-freeness, perfectness, existence of an isolated vertex and many more are all evasive. A major conjecture, attributed to Karp, claims that every nontrivial monotone property, that is a property closed under inserting new edges, is evasive. Its proof when n is a prime power [2] is one of the celebrated applications of topological methods in combinatorics. Unsurprisingly, the known constructions of nonevasive properties are rare and to some extent artificial (see [1, 6] for the original papers and [4, Chapter 3], [3, 2010 Mathematics Subject Classification. 05C99,00A08.