## Abstract

Let P be a graph property. For k ≥ 1, a graph G has property P_{k} iff every induced k-vertex subgraph of G has P. For a graph G we denote by N_{Pk} (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. If P is spanning and k ≥ 1 is fixed, deciding whether a graph G = (V, E) has P_{k} can be done in O(V + E) time. If P is spanning, f_{P}(n) = 0(2^{n3} ), 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^{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 P_{k} 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 (perfect-matching of independent vertices and the Hamiltonian hole are examples of such properties) computing N_{Pk} (G) for graphs G with Δ(G) ≤ d can be done in linear time. If P is an NP-Hard monotone property and ε > 0 is fixed, then P⌊_{ne}⌋ is also NP-Hard. The monotonicity is required as there are NP-Hard properties where P_{k} is easy When k < n.

Original language | English |
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Pages (from-to) | 340-352 |

Number of pages | 13 |

Journal | Journal of Complexity |

Volume | 13 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1997 |

Externally published | Yes |

## ASJC Scopus subject areas

- Algebra and Number Theory
- Statistics and Probability
- Numerical Analysis
- General Mathematics
- Control and Optimization
- Applied Mathematics