How to Eliminate a Graph

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.

[1]  Bruno Courcelle,et al.  The monadic second-order logic of graphs III: tree-decompositions, minor and complexity issues , 1992, RAIRO Theor. Informatics Appl..

[2]  S. Parter.  The Use of Linear Graphs in Gauss Elimination , 1961 .

[3]  Dániel Marx,et al.  Obtaining a Planar Graph by Vertex Deletion , 2007, WG.

[4]  Robert E. Tarjan,et al.  Algorithmic Aspects of Vertex Elimination on Graphs , 1976, SIAM J. Comput..

[5]  D. Karpov.  Spanning trees with many leaves , 2011 .

[6]  Christian Komusiewicz,et al.  Measuring Indifference: Unit Interval Vertex Deletion , 2010, WG.

[7]  Pinar Heggernes,et al.  Minimal triangulations of graphs: A survey , 2006, Discret. Math..

[8]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[9]  J. Pasciak,et al.  Computer solution of large sparse positive definite systems , 1982 .

[10]  Michal Pilipczuk,et al.  Dominating set is fixed parameter tractable in claw-free graphs , 2010, Theor. Comput. Sci..

[11]  Pim van 't Hof,et al.  Parameterized complexity of vertex deletion into perfect graph classes , 2011, Theor. Comput. Sci..

[12]  Leizhen Cai,et al.  Fixed-Parameter Tractability of Graph Modification Problems for Hereditary Properties , 1996, Inf. Process. Lett..

[13]  Geevarghese Philip,et al.  A Quartic Kernel for Pathwidth-One Vertex Deletion , 2010, WG.

[14]  Alan George,et al.  Computer Solution of Large Sparse Positive Definite , 1981 .

[15]  Mihalis Yannakakis,et al.  Edge-Deletion Problems , 1981, SIAM J. Comput..

[16]  R. Sharan,et al.  Complexity classication of some edge modication problems , 1999 .

[17]  Dániel Marx.  Chordal Deletion is Fixed-Parameter Tractable , 2008, Algorithmica.

[18]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[19]  Robin Thomas,et al.  Quickly Excluding a Planar Graph , 1994, J. Comb. Theory, Ser. B.

[20]  Michael R. Fellows,et al.  On problems without polynomial kernels , 2009, J. Comput. Syst. Sci..

[21]  Robin Milner.  An Action Structure for Synchronous pi-Calculus , 1993, FCT.

[22]  Bruce A. Reed,et al.  An (almost) linear time algorithm for odd cycles transversal , 2010, SODA '10.

[23]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[24]  Roded Sharan,et al.  Complexity classification of some edge modification problems , 1999, Discret. Appl. Math..