Reconfiguration on Sparse Graphs

  • Daniel Lokshtanov
  • Amer E. Mouawad
  • Fahad Panolan
  • M. S. Ramanujan
  • Saket Saurabh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9214)

Abstract

A vertex-subset graph problem \(\mathcal {Q}\) defines which subsets of the vertices of an input graph are feasible solutions. A reconfiguration variant of a vertex-subset problem asks, given two feasible solutions \(S_s\) and \(S_t\) of size k, whether it is possible to transform \(S_s\) into \(S_t\) by a sequence of vertex additions and deletions such that each intermediate set is also a feasible solution of size bounded by k. We study reconfiguration variants of two classical vertex-subset problems, namely Independent Set and Dominating Set. We denote the former by \(\textsc {ISR}\) and the latter by \(\textsc {DSR}\). Both \(\textsc {ISR}\) and \(\textsc {DSR}\) are PSPACE-complete on graphs of bounded bandwidth and W[1]-hard parameterized by k on general graphs. We show that \(\textsc {ISR}\) is fixed-parameter tractable parameterized by k when the input graph is of bounded degeneracy or nowhere dense. As a corollary, we answer positively an open question concerning the parameterized complexity of the problem on graphs of bounded treewidth. Moreover, our techniques generalize recent results showing that \(\textsc {ISR}\) is fixed-parameter tractable on planar graphs and graphs of bounded degree. For \(\textsc {DSR}\), we show the problem fixed-parameter tractable parameterized by k when the input graph does not contain large bicliques, a class of graphs which includes degenerate and nowhere dense graphs.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Daniel Lokshtanov
    • 1
  • Amer E. Mouawad
    • 2
  • Fahad Panolan
    • 3
  • M. S. Ramanujan
    • 1
  • Saket Saurabh
    • 3
  1. 1.University of BergenBergenNorway
  2. 2.David R. Cheriton School of Computer ScienceUniversity of WaterlooOntarioCanada
  3. 3.Institute of Mathematical SciencesChennaiIndia

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