### Abstract

We consider the "minor" and "homeomorphic" analogues of the maximum clique problem, i.e., the problems of determining the largest h such that the input graph (on n vertices) has a minor isomorphic to K_{h} or a subgraph homeomorphic to K_{h}, respectively, as well as the problem of finding the corresponding subgraphs. We term them as the maximum clique minor problem and the maximum homeomorphic clique problem, respectively. We observe that a known result of Kostochka and Thomason supplies an O (sqrt(n)) bound on the approximation factor for the maximum clique minor problem achievable in polynomial time. We also provide an independent proof of nearly the same approximation factor with explicit polynomial-time estimation, by exploiting the minor separator theorem of Plotkin et al. Next, we show that another known result of Bollobás and Thomason and of Komlós and Szemerédi provides an O (sqrt(n)) bound on the approximation factor for the maximum homeomorphic clique achievable in polynomial time. On the other hand, we show an Ω (n^{1 / 2 - O (1 / (log n)γ)}) lower bound (for some constant γ, unless NP ⊆ ZPTIME (2^{(log n)O (1)})) on the best approximation factor achievable efficiently for the maximum homeomorphic clique problem, nearly matching our upper bound. Finally, we derive an interesting trade-off between approximability and subexponential time for the problem of subgraph homeomorphism where the guest graph has maximum degree not exceeding three and low treewidth.

Original language | English (US) |
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Pages (from-to) | 149-158 |

Number of pages | 10 |

Journal | Theoretical Computer Science |

Volume | 374 |

Issue number | 1-3 |

DOIs | |

State | Published - Apr 20 2007 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Keywords

- Approximation
- Graph homeomorphism
- Graph minors

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## Cite this

*Theoretical Computer Science*,

*374*(1-3), 149-158. https://doi.org/10.1016/j.tcs.2006.12.021