Evasiveness of Subgraph Containment and Related Properties

Abstract

We prove new results on evasiveness of monotone graph properties by extending the techniques of Kahn, Saks, and Sturtevant [ Combinatorica, 4 (1984), pp. 297--306]. For the property of containing a subgraph isomorphic to a fixed graph, and a fairly large class of related n-vertex graph properties, we show evasiveness for an arithmetic progression of values of n. This implies a $\frac12n^2 - O(n)$ lower bound on the decision tree complexity of these properties. We prove that properties that are preserved under taking graph minors are evasive for all sufficiently large n. This greatly generalizes a theorem due to Best, van Emde Boas, and Lenstra [A Sharpened Version of the Aanderaa--Rosenberg Conjecture, Report ZW 30/74, Mathematisch Centrum, Amsterdam, The Netherlands, 1974] which states that planarity is evasive. We prove a similar result for bipartite subgraph containment.