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## Frequent Subgraph Mining in Outerplanar Graphs (2006)

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### Other Repositories/Bibliography

Venue: | PROC. 12TH ACM SIGKDD INT. CONF. ON KNOWLEDGE DISCOVERY AND DATA MINING |

Citations: | 39 - 7 self |

### Citations

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Citation Context ...luation and finally, Section 6 concludes and discusses some open problems. Due to space limitations, proofs are omitted in this short version. 2 Preliminaries We recall some notions related to graphs =-=[7]-=-. An undirected graph is a pair (V, E), where V = ∅ is a finite set of vertices and E ⊆ {e ⊆ V : |e| = 2} is a set of edges. A labeled undirected graph is a quadruple (V, E, Σ, λ), where (V, E) is an... |

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Citation Context ...orms a forest. For the set of blocks and the forest formed by the bridges of a graph G it holds that their cardinalities are bounded by |V (G)| and they can be enumerated in time O (|V (G)| + |E(G)|) =-=[16]-=-. Let G1 and G2 be graphs. G1 and G2 are isomorphic, denoted G1 ≃ G2, if there is a bijection ϕ : V (G1) → V (G2) such that (i) {u, v} ∈ E(G1) iff {ϕ(u), ϕ(v)} ∈ E(G2), (ii) λG1(u) = λG2(ϕ(u)), (iii) ... |

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Citation Context ...the set of all frequent patterns. Existing approaches to frequent pattern discovery for graphs have therefore resorted to various heuristic strategies and restrictions of the search space (see, e.g., =-=[4, 5, 8, 17]-=-), but have not identified a practically relevant tractable graph class beyond trees. In this paper, we define the class of so called tenuous outerplanar graphs, which is the class of planar graphs th... |

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Citation Context ...neralizes trees, by considering the FTOSM problem we take a step towards going beyond trees in frequent subgraph mining. 4 The Mining Algorithm In this section we present Algorithm 1, an Apriori-like =-=[1]-=- algorithm, that solves the FTOSM problem in incremental polynomial time. For a set D ⊆ Od Σ and integer t ≥ 0, the algorithm computes iteratively the set of t-frequent k-patterns from the set of t-fr... |

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Citation Context ... in such cases it is impossible to enumerate F t Σ,d (D) in time polynomial in the parameters of D. We therefore ask whether the FTOSM problem can be solved in incremental polynomial time (see, e.g., =-=[9]-=-), that is, whether there exists an enumeration algorithm listing the first k elements of F t Σ,d (D) in time polynomial in the combined size of D and the set of these k elements for every k = 1, . . ... |

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Citation Context ...the set of all frequent patterns. Existing approaches to frequent pattern discovery for graphs have therefore resorted to various heuristic strategies and restrictions of the search space (see, e.g., =-=[4, 5, 8, 17]-=-), but have not identified a practically relevant tractable graph class beyond trees. In this paper, we define the class of so called tenuous outerplanar graphs, which is the class of planar graphs th... |

160 |
Computers and Intractability: A Guide to NP-Completeness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...subgraph isomorphic to G is an NP-complete problem. This follows from the fact that outerplanar graphs generalize forests and deciding whether a forest is subgraph isomorphic to a tree is NP-complete =-=[6]-=-. The following stronger negative result is shown in [15]. Theorem 2 Deciding whether a connected outerplanar graph H is subgraph isomorphic to a biconnected outerplanar graph G is NP-complete. If, ho... |

159 | Mining Molecular Fragments: Finding Relevant Substructures of Molecules - Borgelt, Berthold - 2002 |

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139 | Frequent substructure-based approaches for classifying chemical compounds
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Citation Context ...se is one of the central tasks considered in data mining. In addition to be interesting in their own right, frequent patterns can also be used as features for predictive data mining tasks (see, e.g., =-=[5]-=-). For a long time, work on frequent pattern discovery has concentrated on relatively simple notions of patterns and elements in the database as they are typically used for the discovery of associatio... |

98 | Complete mining of frequent patterns from data: Mining graph data
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Citation Context ...the set of all frequent patterns. Existing approaches to frequent pattern discovery for graphs have therefore resorted to various heuristic strategies and restrictions of the search space (see, e.g., =-=[4, 5, 8, 17]-=-), but have not identified a practically relevant tractable graph class beyond trees. In this paper, we define the class of so called tenuous outerplanar graphs, which is the class of planar graphs th... |

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85 |
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Citation Context ...nd count their frequencies. Proposition 1 in Section 2 implies that the number of cycles of a d-tenuous outerplanar graph G is bounded by O (|V (G)|) if d is assumed to be constant. Furthermore, from =-=[13, 16]-=- it follows that the cycles of a graph can be listed with linear delay. Since isomorphism8 Algorithm 2 FREQUENTBICONNECTEDGRAPHS Require: D ⊆ O d Σ for some alphabet Σ and integer d ≥ 0, and integer ... |

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Citation Context ...lt holds [11]. 5 Theorem 4 The problem whether a tree H is subgraph isomorphic to a tree G can be decided in time O ( |V (H)| 1.5 · |V (G)| ) . 5 The bound in Theorem 4 is improved by a log factor in =-=[14]-=-. For the sake of simplicity, we generalize the algorithm in [11] to outerplanar graphs in the long version of this paper. We note that the complexity of our algorithm can also be improved using the i... |

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32 |
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Citation Context ...ph is outerplanar iff each of its blocks is outerplanar [7]. Furthermore, as the blocks of a graph can be computed in linear time [16] and outerplanarity of a block can be decided also in linear time =-=[10, 12]-=-, one can decide in linear time whether a graph is outerplanar. A biconnected outerplanar graph G with n vertices contains at most 2n − 3 edges and has a unique Hamiltonian cycle which bounds the oute... |

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Citation Context ...nd, the frequent graph mining problem is solvable in incremental polynomial time when the graphs in the dataset are restricted to forests and the patterns to trees. This follows e.g. from the results =-=[3]-=-. Since tenuous outerplanar graphs form a practically relevant graph class that naturally generalizes trees, by considering the FTOSM problem we take a step towards going beyond trees in frequent subg... |

19 |
Frequent subtree mining - An overview. Fundamenta Informaticae 66(1-2
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Citation Context ...n which is based on the BB-tree ˜ G of G. By (iii) of Proposition 6, ˜ G is a free tree. Utilizing this property, we can generalize the depth-first canonical representation for free trees (see, e.g., =-=[2]-=-) to outerplanar graphs, and state the following result: Theorem 7 A canonical string representation of a graph in OΣ with n vertices can be computed in time O ( n 2 log n ) . 4.2 Mining Frequent Bico... |

15 | Minimal cycle bases of outerplanar graphs - Leydold, Stadler - 1998 |

14 |
Subgraph isomorphism for biconnected outerplanar graphs in cubic time
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Citation Context ...ph is outerplanar iff each of its blocks is outerplanar [7]. Furthermore, as the blocks of a graph can be computed in linear time [16] and outerplanarity of a block can be decided also in linear time =-=[10, 12]-=-, one can decide in linear time whether a graph is outerplanar. A biconnected outerplanar graph G with n vertices contains at most 2n − 3 edges and has a unique Hamiltonian cycle which bounds the oute... |

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5 |
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Citation Context ... G, H be biconnected outerplanar graphs. Then one can decide in time O ( |V (H)| · |V (G)| 2) whether H is subgraph isomorphic to G. For the special case of trees, the following positive result holds =-=[11]-=-. 5 Theorem 4 The problem whether a tree H is subgraph isomorphic to a tree G can be decided in time O ( |V (H)| 1.5 · |V (G)| ) . 5 The bound in Theorem 4 is improved by a log factor in [14]. For the... |

5 | Maximum Common Subgraph Mining: a Fast and Effective Approach Towards Feature Generation - Schietgat, Costa, et al. |

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1 |
The subgraph isomorphism problem for outerplanar graphs
- Sysło
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(Show Context)
Citation Context ... follows from the fact that outerplanar graphs generalize forests and deciding whether a forest is subgraph isomorphic to a tree is NP-complete [6]. The following stronger negative result is shown in =-=[15]-=-. Theorem 2 Deciding whether a connected outerplanar graph H is subgraph isomorphic to a biconnected outerplanar graph G is NP-complete. If, however, H is also biconnected, the following positive resu... |

1 | conference on data mining (ICDM). IEEE Computer Society, pp 51–58 Calders - Ramon, Van - 1994 |