Contracting Graphs to Split Graphs and Threshold Graphs

We study the parameterized complexity of Split Contraction and Threshold Contraction. In these problems we are given a graph G and an integer k and asked whether G can be modified into a split graph or a threshold graph, respectively, by contracting at most k edges. We present an FPT algorithm for Split Contraction, and prove that Threshold Contraction on split graphs, i.e., contracting an input split graph to a threshold graph, is FPT when parameterized by the number of contractions. To give a complete picture, we show that these two problems admit no polynomial kernels unless NP\subseteq coNP/poly.

[1]  Michael R. Fellows,et al.  On Problems without Polynomial Kernels (Extended Abstract) , 2008, ICALP.

[2]  Jianer Chen,et al.  A 2k kernel for the cluster editing problem , 2012, J. Comput. Syst. Sci..

[3]  Petr A. Golovach,et al.  Increasing the minimum degree of a graph by contractions , 2011, Theor. Comput. Sci..

[4]  Saket Saurabh,et al.  Incompressibility through Colors and IDs , 2009, ICALP.

[5]  Jiong Guo,et al.  Problem Kernels for NP-Complete Edge Deletion Problems: Split and Related Graphs , 2007, ISAAC.

[6]  Uriel Feige,et al.  Approximating the domatic number , 2000, STOC '00.

[7]  Pim van 't Hof,et al.  Obtaining a Bipartite Graph by Contracting Few Edges , 2011, FSTTCS.

[8]  Pim van 't Hof,et al.  Contracting Graphs to Paths and Trees , 2011, Algorithmica.

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

[10]  Saket Saurabh,et al.  On the Hardness of Eliminating Small Induced Subgraphs by Contracting Edges , 2013, IPEC.

[11]  Petr A. Golovach,et al.  Obtaining planarity by contracting few edges , 2013, Theor. Comput. Sci..

[12]  Leizhen Cai,et al.  Contracting Few Edges to Remove Forbidden Induced Subgraphs , 2013, IPEC.

[13]  Pim van 't Hof,et al.  Edge contractions in subclasses of chordal graphs , 2011, Discret. Appl. Math..

[14]  Tao Jiang,et al.  Phylogenetic k-Root and Steiner k-Root , 2000, ISAAC.

[15]  Anders Yeo,et al.  Kernel Bounds for Disjoint Cycles and Disjoint Paths , 2009, ESA.

[16]  Fahad Panolan,et al.  Faster Parameterized Algorithms for Deletion to Split Graphs , 2012, SWAT.