## Abstract

This paper investigates a graph enumeration problem, called the maximalP -subgraphs problem, where P is a hereditary or connected-hereditary graph property. Formally, given a graph G, the maximal P-subgraphs problem is to generate all maximal induced subgraphs of G that satisfy P. This problem differs from the well-known node-deletion problem, studied by Yannakakis and Lewis [J. Lewis, On the complexity of the maximum subgraph problem, in: Proc. 10th Annual ACM Symposium on Theory of Computing, ACM Press, New York, USA, 1978, pp. 265-274; M. Yannakakis, Node- and edge-deletion NP-complete problems, in: Proc. 10th Annual ACM Symposium on Theory of Computing, ACM Press, New York, USA, 1978, pp. 253-264; J. Lewis, M. Yannakakis, The node-deletion problem for hereditary properties is NP-complete, J. Comput. System Sci. 20 (2) (1980) 219-230]. In the maximal P-subgraphs problem, the goal is to produce all (locally) maximal subgraphs of a graph that have property P, whereas in the node-deletion problem, the goal is to find a single (globally) maximum size subgraph with property P. Algorithms are presented that reduce the maximal P-subgraphs problem to an input-restricted version of this problem. These algorithms imply that when attempting to efficiently solve the maximal P-subgraphs problem for a specific P, it is sufficient to solve the restricted case. The main contributions of this paper are characterizations of when the maximal P-subgraphs problem is in a complexity class C (e.g., polynomial delay, total polynomial time).

Original language | American English |
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Pages (from-to) | 1147-1159 |

Number of pages | 13 |

Journal | Journal of Computer and System Sciences |

Volume | 74 |

Issue number | 7 |

DOIs | |

State | Published - Nov 2008 |

### Bibliographical note

Funding Information:E-mail addresses: sara@cs.huji.ac.il (S. Cohen), bennyk@cs.huji.ac.il (B. Kimelfeld), sagiv@cs.huji.ac.il (Y. Sagiv). 1 Sara Cohen was partially supported by The Israel Science Foundation (Grant 1032/05). 2 Benny Kimelfeld and Yehoshua Sagiv were partially supported by The Israel Science Foundation (Grant 893/05).

## Keywords

- Complexity classes
- Enumeration
- Graph properties
- Hereditary properties
- Maximal subgraphs