Recognizing Global Occurrence of Local Properties

Abstract

Let $\text{P}$ be a graph property. Fork≥ 1, a graphGhas propertyP_kiff every inducedk-vertex subgraph ofGhas $\text{P}$. For a graphGwe denote byN_Pk(G) the number of inducedk-vertex subgraphs ofGhaving $\text{P}$. 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 $\text{P}$ is recursive if one can determine if a graphGonnvertices has $\text{P}$ in timeO(f_P(n)) wheref_P(n) is some recursive function ofn. We consider only recursive properties. Our main results are the following.

• If $\text{P}$ is spanning andk≥ 1 is fixed, deciding whether a graphG= (V,E) hasP_kcan be done inO(V+E) time.

• If $\text{P}$ is spanning,f_P(n) = O(2ⁿ³), andk=O((logn/log logn)^1/3), deciding whetherGhasP_kcan be done in polynomial time. Furthermore, if $\text{P}$ is a monotone-increasing simple property withf_P(n) =O(2ⁿ²) (Hamiltonicity, perfect-matching, ands-connectivity are just a few examples of such properties) andk=O($\text{logn/log logn}$), deciding whetherGhasP_kcan be done in polynomial time.

• Ifk≥ 1 andd≥ 1 are fixed, and $\text{P}$ 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) computingN_Pk(G) for graphsGwith Δ(G) ≤dcan be done in linear time.

• If $\text{P}$ is an NP-Hard monotone property and ε > 0 is fixed, thenP_⌊n^ε⌋is also NP-Hard. The monotonicity is required as there are NP-Hard properties whereP_kis easy whenk<n.