How to eliminate a graph

Golovach, P.A.; Heggernes, P.; van 't Hof, P.; Manne, F.; Paulusma, D.; Pilipczuk, M.

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.

Citation

Golovach, P., Heggernes, P., van 't Hof, P., Manne, F., Paulusma, D., & Pilipczuk, M. (2012, December). How to eliminate a graph

• Presentation Conference Type: Conference Paper (published)
• Publication Date: 2012
• Deposit Date: Mar 11, 2013
• Print ISSN: 0302-9743
• Pages: 320-331
• Series Title: Lecture notes in computer science
• Series Number: 7551
• Series ISSN: 0302-9743,1611-3349
• Book Title: Graph-theoretic concepts in computer science: 38th international workshop, WG 2012, Jerusalem, Israel, June 26-28, 2012, revised selected papers.
• ISBN: 9783642346101
• DOI: https://doi.org/10.1007/978-3-642-34611-8_32
• Public URL: https://durham-repository.worktribe.com/output/1156186
• Additional Information: Series: Lecture Notes in Computer Science, Volume 7551