Recognizing Global Occurrence of Local Properties

Let P be a graph property. Fork? 1, a graphGhas propertyPkiff every inducedk-vertex subgraph ofGhas P. For a graphGwe denote byNPk(G) the number of inducedk-vertex subgraphs ofGhaving P. A property is calledspanningif it does not hold for graphs that contain isolated vertices. A property is calledconnectedif it does not hold for graphs with more than one connected component. Many familiar graph properties are spanning or connected. We also define the notion ofsimpleproperties which also applies to many well-known monotone graph properties. A property P is recursive if one can determine if a graphGonnvertices has P in timeO(fP(n)) wherefP(n) is some recursive function ofn. We consider only recursive properties. Our main results are the following.?If P is spanning andk? 1 is fixed, deciding whether a graphG= (V,E) hasPkcan be done inO(V+E) time.?If P is spanning,fP(n) = O(2n3), andk=O((logn/log logn)1/3), deciding whetherGhasPkcan be done in polynomial time. Furthermore, if P is a monotone-increasing simple property withfP(n) =O(2n2) (Hamiltonicity, perfect-matching, ands-connectivity are just a few examples of such properties) andk=O(logn/log logn), deciding whetherGhasPkcan be done in polynomial time.?Ifk? 1 andd? 1 are fixed, and P is either a connected property (Hamiltonicity is an example of such a property) or a monotone-decreasing infinitely-simple property (perfect-matching of independent vertices and the Hamiltonian hole are examples of such properties) computingNPk(G) for graphsGwith ?(G) ?dcan be done in linear time.?If P is an NP-Hard monotone property and ? > 0 is fixed, thenP?n??is also NP-Hard. The monotonicity is required as there are NP-Hard properties wherePkis easy whenk