Parameterized Domination in Circle Graphs

A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Appl. Math., 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 G has a dominating set inducing a graph isomorphic to T is NP-complete when T is in the input, and FPT when parameterized by t=|V(T)|. We prove that the FPT algorithm runs in subexponential time, namely $2^{\mathcal{O}(t \cdot\frac{\log\log t}{\log t})} \cdot n^{\mathcal{O}(1)}$, where n=|V(G)|.

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

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

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

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

[5]  Peter Rossmanith,et al.  Dynamic Programming on Tree Decompositions Using Generalised Fast Subset Convolution , 2009, ESA.

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

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

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

[9]  Shlomo Moran,et al.  Non Deterministic Polynomial Optimization Problems and their Approximations , 1977, Theor. Comput. Sci..

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

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

[12]  Ehab S. Elmallah,et al.  Polygon Graph Recognition , 1998, J. Algorithms.

[13]  Ton Kloks,et al.  Treewidth of Circle Graphs , 1993, ISAAC.

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

[15]  Dieter Kratsch,et al.  Exponential time algorithms for the minimum dominating set problem on some graph classes , 2006, TALG.

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

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

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

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

[20]  Erik Jan van Leeuwen,et al.  Domination When the Stars Are Out , 2010, ICALP.

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

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

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

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

[25]  Fabrizio Grandoni,et al.  Resilient dictionaries , 2009, TALG.

[26]  Krzysztof Pietrzak,et al.  On the parameterized complexity of the fixed alphabet shortest common supersequence and longest common subsequence problems , 2003, J. Comput. Syst. Sci..

[27]  Philippe Flajolet,et al.  Analytic Combinatorics , 2009 .

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

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

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

[31]  P. Flajolet,et al.  Analytic Combinatorics: RANDOM STRUCTURES , 2009 .

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

[33]  Christophe Paul,et al.  Parameterized Domination in Circle Graphs , 2012, WG.

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

[35]  Dimitrios M. Thilikos,et al.  Invitation to fixed-parameter algorithms , 2007, Comput. Sci. Rev..

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

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

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

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

[40]  Dimitrios M. Thilikos,et al.  Dynamic programming for graphs on surfaces , 2014, TALG.

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

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

[43]  Stéphane Pérennes,et al.  Degree-Constrained Subgraph Problems: Hardness and Approximation Results , 2008, WAOA.