### Abstract

Vertex elimination is a graph operation that turns the neighborhood of a vertex into a clique and removes the vertex itself. It has widely known applications within sparse matrix computations. We define the ELIMINATION problem as follows: given two graphs G and H, decide whether H can be obtained from G by |V(G)|−|V(H)| vertex eliminations. We show that ELIMINATION is W[1]-hard when parameterized by |V(H)|, even if both input graphs are split graphs, and W[2]-hard when parameterized by |V(G)|−|V(H)|, even if H is a complete graph. On the positive side, we show that ELIMINATION admits a kernel with at most 5|V(H)| vertices in the case when G is connected and H is a complete graph, which is in sharp contrast to the W[1]-hardness of the related CLIQUE problem. We also study the case when either G or H is tree. The computational complexity of the problem depends on which graph is assumed to be a tree: we show that ELIMINATION can be solved in polynomial time when H is a tree, whereas it remains NP-complete when G is a tree.

Original language | English |
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Pages (from-to) | 99-125 |

Journal | Algorithmica |

Volume | 72 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 |

Externally published | Yes |

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## Cite this

Golovach, P. A., Heggernes, P., van 't Hof, P., Manne, F., Paulusma, D., & Pilipczuk, M. (2015). Modifying a Graph Using Vertex Elimination.

*Algorithmica*,*72*(1), 99-125. https://doi.org/10.1007/s00453-013-9848-2