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. © 2012 Springer-Verlag Berlin Heidelberg.
CITATION STYLE
Golovach, P. A., Heggernes, P., Van’T Hof, P., Manne, F., Paulusma, D., & Pilipczuk, M. (2012). How to eliminate a graph. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 7551 LNCS, pp. 320–331). https://doi.org/10.1007/978-3-642-34611-8_32
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