How to eliminate a graph

  • Authors:

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

  • Affiliations:

  • School of Engineering and Computing Sciences, Durham University, UK;Department of Informatics, University of Bergen, Norway;Department of Informatics, University of Bergen, Norway;Department of Informatics, University of Bergen, Norway;School of Engineering and Computing Sciences, Durham University, UK;Department of Informatics, University of Bergen, Norway

  • Venue:
  • WG'12 Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science
  • Year:
  • 2012

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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 study the parameterized complexity of the Elimination problem. 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.