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On the Computational Complexity of Graph Problems

Ahadi, Arash | 2011

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 42292 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Jafari, Amir
  7. Abstract:
  8. Computational complexity of graph problems is an important branch in the-oretical computer science. We introduce to some of ideas for computing the complexity of graph problems with some kind and beautiful examples. Next, we show hardness and inapproximability of some problems. Representation number of graphs has been introduce by Pavel Erdos by Number theory. We prove n1−ϵ inapproximability of that. Lucky number η has been studied by Grytczuk et.al . We show for planar and 3-colorable graphs, it is NP-Complete whether η = 2. Note that since a conjecture, for those graphs, 2 ≤ η ≤ 3. Also for each k ≥ 2, we show NP-completeness of η ≤ k for the graphs. Proper orientation number −→ is a concept that makes an adjacency between the orientation and the coloring. For all graphs, χ − 1 ≤ χ ≤ ∆; we show deciding the equality of both side is NP-com. Also we show that for planars deciding −→ = 2 is NP-com. This problem is trivial for regulars; For those we show the hardness of −→ = 3. Proper domination number is a bridge between the concepts of the domination and the coloring. About the inapproximability of it, we present some results for some classes of graphs
  9. Keywords:
  10. Computational Complexity ; Non-Approximation ; Approximate Algorithm ; Graphs Representation ; Lucky Labeling ; Proper Orientation ; Planar Graph ; Dominating Set

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