Graph Properties in Node-Query Setting: Effect of Breaking Symmetry

The query complexity of graph properties is well-studied when queries are on edges. We investigate the same when queries are on nodes. In this setting a graphG = (V ,E)on n vertices and a propertyP are given. A blackbox access to an unknown subset S ⊆V is provided via queries of the form ‘Does i ∈ S?’. We are interested in the minimum number of queries needed in worst case in order to determine whether G[S] – the subgraph of G induced on S – satisfies P . Apart from being combinatorially rich, this setting appears to be a natural abstraction of several scenarios in areas including computer networks and social networkswhere one is interested in properties of the underlying sub-network on a set of active nodes. Another reason why we found this setting interesting is because it allows us to initiate a systematic study of breaking symmetry in the context of query complexity of graph properties. In particular, we focus on hereditary graph properties – the properties closed under deletion of vertices as well as edges. The famous Evasiveness Conjecture asserts that even with a minimal symmetry assumption on G, namely that of vertex-transitivity, the query complexity for any hereditary graph property in our setting is the worst possible, i.e., n. We show that in the absence of any symmetry on G it can fall as low as O(n1/(d+1)) where d denotes the minimum possible degree of a minimal forbidden sub-graph for P . In particular, every hereditary property benefits at least quadratically. The main question left open is: can it go exponentially low for some hereditary property? We show that the answer is no for any hereditary property with finitely many forbidden subgraphs by exhibiting a bound of Ω(n1/k) for some constant k depending only on the property. For general ones we rule out the possibility of the query complexity falling down to constant by showing Ω(logn/ log logn) bound. Interestingly, our lower bound proofs rely on the famous Sunflower Lemma due to Erdös and Rado.