Recognizing Global Occurrence of Local Properties

Let P be a graph property. For k ≥ 1, a graph G has property Pk iff every induced k-vertex subgraph of G has P. For a graph G we denote by NPk(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(fP (n)) where fP (n) is some recursive function of n. We consider only recursive properties. Our main results are the following. • If P is spanning and k ≥ 1 is fixed, deciding whether a graph G = (V,E) has Pk can be done in O(V + E) time. • If P is spanning, fP (n) = O(2 3 ) and k = O((log n/ log log n)), deciding whether G has Pk can be done in polynomial time. Furthermore, if P is a monotone-increasing simple property with fP (n) = O(2 n2) (Hamiltonicity, perfect-matching and s-connectivity are just a few examples of such properties) and k = O( √ log n/log log n), deciding whether G has Pk can be done in polynomial time. • If k ≥ 1 and d ≥ 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 (perfectmatching of independent vertices and Hamiltonian hole are examples of such properties) computing NPk(G) for graphs G with ∆(G) ≤ d can be done in linear time. • If P is an NP-Hard monotone property and > 0 is fixed, Then Pbn c is also NP-Hard. The monotonicity is required as there are NP-Hard properties where Pk is easy when k < n.