### Abstract

The graph inference from a walk for a class C of undirected edgecolored graphs is, given a string x of colors, finding the smallest graph G in C that allows a traverse of all edges in G whose sequence of edge-colors is x, called a walk for x. We prove that the graph inference from a walk for trees of bounded degree k is NP-complete for any k ≥ 3, while the problem for trees without any degree bound constraint is known to be solvable in O(n) time, where n is the length of the string. Furthermore, the problem for a special class of trees of bounded degree 3, called (1,1)-caterpillars, is shown to be NP-complete. This contrast with the case that the problem for linear chains is known to be solvable in O(nlog n) time since a (1,1)-caterpillar is obtained by attaching at most one hair of length one to each node of a linear chain. We also show the MAXSNP-hardness of these problems.

Original language | English |
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Title of host publication | Mathematical Foundations of Computer Science 1995 - 20th International Symposium, MFCS 1995, Proceedings |

Editors | Jiri Wiedermann, Petr Hajek |

Publisher | Springer Verlag |

Pages | 257-266 |

Number of pages | 10 |

ISBN (Print) | 3540602461, 9783540602460 |

Publication status | Published - Jan 1 1995 |

Event | 20th International Symposium on Mathematical Foundations of Computer Science, MFCS 1995 - Prague, Czech Republic Duration: Aug 28 1995 → Sep 1 1995 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 969 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 20th International Symposium on Mathematical Foundations of Computer Science, MFCS 1995 |
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Country | Czech Republic |

City | Prague |

Period | 8/28/95 → 9/1/95 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Mathematical Foundations of Computer Science 1995 - 20th International Symposium, MFCS 1995, Proceedings*(pp. 257-266). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 969). Springer Verlag.

**Graph inference from a walk for trees of bounded degree 3 is NP-complete.** / Maruyama, Osamu; Miyano, Satoru.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Mathematical Foundations of Computer Science 1995 - 20th International Symposium, MFCS 1995, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 969, Springer Verlag, pp. 257-266, 20th International Symposium on Mathematical Foundations of Computer Science, MFCS 1995, Prague, Czech Republic, 8/28/95.

}

TY - GEN

T1 - Graph inference from a walk for trees of bounded degree 3 is NP-complete

AU - Maruyama, Osamu

AU - Miyano, Satoru

PY - 1995/1/1

Y1 - 1995/1/1

N2 - The graph inference from a walk for a class C of undirected edgecolored graphs is, given a string x of colors, finding the smallest graph G in C that allows a traverse of all edges in G whose sequence of edge-colors is x, called a walk for x. We prove that the graph inference from a walk for trees of bounded degree k is NP-complete for any k ≥ 3, while the problem for trees without any degree bound constraint is known to be solvable in O(n) time, where n is the length of the string. Furthermore, the problem for a special class of trees of bounded degree 3, called (1,1)-caterpillars, is shown to be NP-complete. This contrast with the case that the problem for linear chains is known to be solvable in O(nlog n) time since a (1,1)-caterpillar is obtained by attaching at most one hair of length one to each node of a linear chain. We also show the MAXSNP-hardness of these problems.

AB - The graph inference from a walk for a class C of undirected edgecolored graphs is, given a string x of colors, finding the smallest graph G in C that allows a traverse of all edges in G whose sequence of edge-colors is x, called a walk for x. We prove that the graph inference from a walk for trees of bounded degree k is NP-complete for any k ≥ 3, while the problem for trees without any degree bound constraint is known to be solvable in O(n) time, where n is the length of the string. Furthermore, the problem for a special class of trees of bounded degree 3, called (1,1)-caterpillars, is shown to be NP-complete. This contrast with the case that the problem for linear chains is known to be solvable in O(nlog n) time since a (1,1)-caterpillar is obtained by attaching at most one hair of length one to each node of a linear chain. We also show the MAXSNP-hardness of these problems.

UR - http://www.scopus.com/inward/record.url?scp=84947918349&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84947918349&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84947918349

SN - 3540602461

SN - 9783540602460

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 257

EP - 266

BT - Mathematical Foundations of Computer Science 1995 - 20th International Symposium, MFCS 1995, Proceedings

A2 - Wiedermann, Jiri

A2 - Hajek, Petr

PB - Springer Verlag

ER -