Algorithmica

, Volume 72, Issue 1, pp 99–125

# Modifying a Graph Using Vertex Elimination

• Petr A. Golovach
• Pinar Heggernes
• Pim van ’t Hof
• Fredrik Manne
• Daniël Paulusma
• Michał Pilipczuk
Article

## 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 $$\mathsf {W[1]}$$-hard when parameterized by |V(H)|, even if both input graphs are split graphs, and $$\mathsf {W[2]}$$-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 $$\mathsf {W[1]}$$-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.

## Keywords

Graph modification problems Vertex elimination Parameterized complexity Linear kernel

## Notes

### Acknowledgements

We would like to thank Łukasz Kowalik for an inspiring discussion on the theorem of Kleitman and West. We also thank the two anonymous referees for their useful comments and suggestions that helped us to improve the presentation of our paper.

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## Authors and Affiliations

• Petr A. Golovach
• 1
• Pinar Heggernes
• 1
• Pim van ’t Hof
• 1
Email author
• Fredrik Manne
• 1
• Daniël Paulusma
• 2
• Michał Pilipczuk
• 1
1. 1.Department of InformaticsUniversity of BergenBergenNorway
2. 2.School of Engineering and Computing SciencesDurham UniversityDurhamUK