Complexity of finding embeddings in a k-tree.

*(English)*Zbl 0611.05022A k-tree is an undirected graph that can be reduced to the k-complete graph by a sequence of removals of k-degree vertices with completely connected neighbors. A partial k-tree is a subgraph of a k-tree; \(k_ t(G)\) is the smallest k for which G is a partial k-tree.

The problem PARTIAL K-TREE is: given a graph G and an integer k, is \(k_ t(G)\leq k?\) The authors prove that PARTIAL K-TREE problem is NP-complete. For a fixed k, they present a polynomial time algorithm for that problem which, unfortunately, is of degree k.

The problem PARTIAL K-TREE is: given a graph G and an integer k, is \(k_ t(G)\leq k?\) The authors prove that PARTIAL K-TREE problem is NP-complete. For a fixed k, they present a polynomial time algorithm for that problem which, unfortunately, is of degree k.

Reviewer: G.Slutzki

##### MSC:

05C10 | Planar graphs; geometric and topological aspects of graph theory |

68Q25 | Analysis of algorithms and problem complexity |

90C39 | Dynamic programming |

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\textit{S. Arnborg} et al., SIAM J. Algebraic Discrete Methods 8, 277--284 (1987; Zbl 0611.05022)

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##### References:

[1] | Arnborg, S., Reduced state enumeration—another algorithm for reliability evaluation, IEEE Trans. Reliability, R-27, 101, (1978) · Zbl 0436.60062 |

[2] | Arnborg, Stefan, Efficient algorithms for combinatorial problems on graphs with bounded decomposability—a survey, BIT, 25, 2, (1985) · Zbl 0573.68018 |

[3] | Arnborg, Stefan; Proskurowski, Andrzej, Characterization and recognition of partial 3-trees, SIAM J. Algebraic Discrete Methods, 7, 305, (1986) · Zbl 0597.05027 |

[4] | Linear time algorithms for NP-hard problems on graphs embedded in k-treesTRITA-NA-8404The Royal Institute of Technology1984 |

[5] | Bondy, J. A.; Murty, U. S. R., Graph theory with applications, (1976) · Zbl 1226.05083 |

[6] | Colbourn, CharlesJ.; Proskurowski, Andrzej, Concurrent transmissions in broadcast networks, Automata, languages and programming (Antwerp, 1984), 172, 128, (1984), Springer, Berlin · Zbl 0554.94021 |

[7] | Corneil, D. G.; Keil, J. M., A dynamic programming approach to the dominating set problem on k-trees, SIAM J. Algebraic Discrete Methods, 8, 535, (1987) · Zbl 0635.05040 |

[8] | Farley, ArthurM., Networks immune to isolated failures, Networks, 11, 255, (1981) · Zbl 0459.94036 |

[9] | Farley, ArthurM.; Proskurowski, Andrzej, Networks immune to isolated line failures, Networks, 12, 393, (1982) · Zbl 0493.94020 |

[10] | Garey, MichaelR.; Johnson, DavidS., Computers and intractability, (1979) · Zbl 0411.68039 |

[11] | Gilmore, P. C.; Hoffman, A. J., A characterization of comparability graphs and of interval graphs, Canad. J. Math., 16, 539, (1964) · Zbl 0121.26003 |

[12] | The most reliable series parallel networksTR 83-7Dept. of Computing Science, University of Saskatchewan1983 |

[13] | Proskurowski, Andrzej, Separating subgraphs in k-trees: cables and caterpillars, Discrete Math., 49, 275, (1984) · Zbl 0543.05041 |

[14] | Rose, DonaldJ., Triangulated graphs and the elimination process, J. Math. Anal. Appl., 32, 597, (1970) · Zbl 0216.02602 |

[15] | Rose, DonaldJ., On simple characterizations of k-trees, Discrete Math., 7, 317, (1974) · Zbl 0285.05128 |

[16] | Wald, A.; Colbourn, C. J., Steiner trees, partial 2-trees, and minimum IFI networks, Networks, 13, 159, (1983) · Zbl 0529.68036 |

[17] | Yannakakis, Mihalis, Computing the minimum fill-in is NP-complete, SIAM J. Algebraic Discrete Methods, 2, 77, (1981) · Zbl 0496.68033 |

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