NP-completeness results for some problems on subclasses of bipartite and chordal graphs

  • Authors:

  • Katerina Asdre;Stavros D. Nikolopoulos

  • Affiliations:

  • Department of Computer Science, University of Ioannina, P.O. Box 1186, GR-45110 Ioannina, Greece;Department of Computer Science, University of Ioannina, P.O. Box 1186, GR-45110 Ioannina, Greece

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2007

Quantified Score

Hi-index 5.25

Visualization

Abstract

Extending previous NP-completeness results for the harmonious coloring problem and the pair-complete coloring problem on trees, bipartite graphs and cographs, we prove that these problems are also NP-complete on connected bipartite permutation graphs. We also study the k-path partition problem and, motivated by a recent work of Steiner [G. Steiner, On the k-path partition of graphs, Theoret. Comput. Sci. 290 (2003) 2147-2155], where he left the problem open for the class of convex graphs, we prove that the k-path partition problem is NP-complete on convex graphs. Moreover, we study the complexity of these problems on two well-known subclasses of chordal graphs namely quasi-threshold and threshold graphs. Based on the work of Bodlaender [H.L. Bodlaender, Achromatic number is NP-complete for cographs and interval graphs, Inform. Process. Lett. 31 (1989) 135-138], we show NP-completeness results for the pair-complete coloring and harmonious coloring problems on quasi-threshold graphs. Concerning the k-path partition problem, we prove that it is also NP-complete on this class of graphs. It is known that both the harmonious coloring problem and the k-path partition problem are polynomially solvable on threshold graphs. We show that the pair-complete coloring problem is also polynomially solvable on threshold graphs by describing a linear-time algorithm.