Data reduction for graph coloring problems

  • Authors:

  • Bart M. P. Jansen;Stefan Kratsch

  • Affiliations:

  • Utrecht University, P.O. Box 80.089, 3508 TB Utrecht, The Netherlands;Technical University Berlin, Germany

  • Venue:
  • Information and Computation
  • Year:
  • 2013

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Abstract

This paper studies the kernelization complexity of graph coloring problems with respect to certain structural parameterizations of the input instances. We are interested in how well polynomial-time data reduction can provably shrink instances of coloring problems, in terms of the chosen parameter. It is well known that deciding 3-colorability is already NP-complete, hence parameterizing by the requested number of colors is not fruitful. Instead, we pick up on a research thread initiated by Cai (DAM, 2003) who studied coloring problems parameterized by the modification distance of the input graph to a graph class on which coloring is polynomial-time solvable; for example parameterizing by the number k of vertex-deletions needed to make the graph chordal. We obtain various upper and lower bounds for kernels of such parameterizations of q-Coloring, complementing Cai@?s study of the time complexity with respect to these parameters. Our results show that the existence of polynomial kernels for q-Coloring parameterized by the vertex-deletion distance to a graph class F is strongly related to the existence of a function f(q) which bounds the number of vertices which are needed to preserve the no-answer to an instance of q-List Coloring on F.