Parameterized Domination in Circle Graphs

A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Applied Mathematics, 42(1):51-63, 1993] proved that Dominating Set, Connected Dominating Set, and Total Dominating Set are NP-complete in circle graphs. To the best of our knowledge, nothing was known about the parameterized complexity of these problems in circle graphs. In this paper we prove the following results, which contribute in this direction: · Dominating Set, Independent Dominating Set, Connected Dominating Set, Total Dominating Set, and Acyclic Dominating Set are W[1]-hard in circle graphs, parameterized by the size of the solution. · Whereas both Connected Dominating Set and Acyclic Dominating Set are W[1]-hard in circle graphs, it turns out that Connected Acyclic Dominating Set is polynomial-time solvable in circle graphs. · If T is a given tree, deciding whether a circle graph has a dominating set isomorphic to T is NP-complete when T is in the input, and FPT when parameterized by |V(T)|. We prove that the FPT algorithm is subexponential.

[1]  Ton Kloks.  Treewidth of Circle Graphs , 1996, Int. J. Found. Comput. Sci..

[2]  Ehab S. Elmallah,et al.  Independence and domination in Polygon Graphs , 1993, Discret. Appl. Math..

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

[4]  Rolf Niedermeier,et al.  Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs , 2002, Algorithmica.

[5]  Guangjun Xu,et al.  Acyclic domination on bipartite permutation graphs , 2006, Inf. Process. Lett..

[6]  Rolf Niedermeier,et al.  Invitation to Fixed-Parameter Algorithms , 2006 .

[7]  Walter Unger,et al.  On the k-Colouring of Circle-Graphs , 1988, STACS.

[8]  Jörg Flum,et al.  Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series) , 2006 .

[9]  Erik Jan van Leeuwen,et al.  k-Gap Interval Graphs , 2011, LATIN.

[10]  A. Itai,et al.  QUEUES, STACKS AND GRAPHS , 1971 .

[11]  Fanica Gavril,et al.  Algorithms for a maximum clique and a maximum independent set of a circle graph , 1973, Networks.

[12]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[13]  Dániel Marx,et al.  Parameterized Complexity of Independence and Domination on Geometric Graphs , 2006, IWPEC.

[14]  Noga Alon,et al.  Kernels for the Dominating Set Problem on Graphs with an Excluded Minor , 2008, Electron. Colloquium Comput. Complex..

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

[16]  Jörg Flum,et al.  Parameterized Complexity Theory , 2006, Texts in Theoretical Computer Science. An EATCS Series.

[17]  Douglas F. Rall,et al.  Acyclic domination , 2000, Discret. Math..

[18]  Michael R. Fellows,et al.  On the parameterized complexity of multiple-interval graph problems , 2009, Theor. Comput. Sci..

[19]  J. Mark Keil.  The Complexity of Domination Problems in Circle Graphs , 1993, Discret. Appl. Math..

[20]  Naveed A. Sherwani,et al.  Algorithms for VLSI Physical Design Automation , 1999, Springer US.

[21]  Lorna Stewart,et al.  Approximating the minimum clique cover and other hard problems in subtree filament graphs , 2006, Discret. Appl. Math..

[22]  Walter Unger,et al.  The Complexity of Colouring Circle Graphs (Extended Abstract) , 1992, STACS.

[23]  Peter Damaschke,et al.  The Hamiltonian Circuit Problem for Circle Graphs is NP-Complete , 1989, Inf. Process. Lett..

[24]  Jeremy P. Spinrad,et al.  Recognition of Circle Graphs , 1994, J. Algorithms.

[25]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs: Definable Sets of Finite Graphs , 1988, WG.

[26]  Emeric Gioan,et al.  Circle Graph Recognition in Time O(n+m) α(n+m) , 2011, ArXiv.

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

[28]  Fanica Gavril,et al.  Minimum weight feedback vertex sets in circle graphs , 2008, Inf. Process. Lett..

[29]  Sriram V. Pemmaraju,et al.  Hardness of Approximating Independent Domination in Circle Graphs , 1999, ISAAC.

[30]  Yong Zhang,et al.  Parameterized Complexity in Multiple-Interval Graphs: Domination , 2011, IPEC.