Let P be a graph property. For k β₯ 1, a graph G has property P k iff every induced kvertex subgraph of G has P. For a graph G we denote by N P k (G) the number of induced k-vertex subgraphs of G having P. A property is called spanning if it does not hold for graphs that contain isolated vertices. A property is called connected if 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 of simple properties which also applies to many well-known monotone graph properties. A property P is recursive if one can determine if a graph G on n vertices has P in time O( f P (n)) where f P (n) is some recursive function of n. We consider only recursive properties. Our main results are the following.
β’ If P is spanning and k β₯ 1 is fixed, deciding whether a graph G = (V, E) has P k can be done in O(V + E) time.
β’ If P is spanning, f P (n) = O(2 n 3 ), and k = O((log n/log log n) 1/3 ), deciding whether G has P k can be done in polynomial time. Furthermore, if P is a monotoneincreasing simple property with f P (n) = O(2 n 2 ) (Hamiltonicity, perfect-matching, and s-connectivity are just a few examples of such properties) and k = O( β log n/ log log n ), deciding whether G has P k can be done in polynomial time.
β’ If k β₯ 1 and d β₯ 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) computing N P k (G) for graphs G with 1(G) β€ d can be done in linear time.
β’ If P is an NP-Hard monotone property and Ξ΅ > 0 is fixed, then P n Ξ΅ is also NP-Hard. The monotonicity is required as there are NP-Hard properties where P k is easy when k < n.