WALCOM 2012: WALCOM: Algorithms and Computation pp 17-27

# Generalized Above Guarantee Vertex Cover and r-Partization

• R. Krithika
• N. S. Narayanaswamy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7157)

## Abstract

Vertex cover and odd cycle transversal are minimum cardinality sets of vertices of a graph whose deletion makes the resultant graph 1-colorable and 2-colorable, respectively. As a natural generalization of these well-studied problems, we consider the Graph r-Partization problem of finding a minimum cardinality set of vertices whose deletion makes the graph r-colorable. We explore further connections to Vertex Cover by introducing Generalized Above Guarantee Vertex Cover, a variant of Vertex Cover defined as: Given a graph G, a clique cover $$\cal K$$ of G and a non-negative integer k, does G have a vertex cover of size at most $$k+\sum_{C \in \cal K}(|C|-1)$$? We study the parameterized complexity hardness of this problem by a reduction from r-Partization. We then describe sequacious fixed-parameter tractability results for r-Partization, parameterized by the solution size k and the required chromaticity r, in perfect graphs and split graphs. For Odd Cycle Transversal, we describe an O *(2 k ) algorithm for perfect graphs and a polynomial-time algorithm for co-chordal graphs.

## Keywords

Parameterized complexity Generalized above guarantee vertex cover Odd cycle transversal r-Partization Perfect graphs Split graphs

## References

1. 1.
Courcelle, B.: The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation 85, 12–75 (1990)
2. 2.
Reed, B.A., Smith, K., Vetta, A.: Finding odd cycle transversals. Operations Research Letters 32, 299–301 (2004)
3. 3.
Jansen, B.M.P., Kratsch, S.: On polynomial kernels for structural parameterizations of odd cycle transversal. To appear in Proceedings of IPEC 2011 (2011)Google Scholar
4. 4.
West, D.B.: Introduction to graph theory. Prentice Hall of India (2003)Google Scholar
5. 5.
Corneil, D.G., Fonlupt, J.: The complexity of generalized clique covering. Discrete Applied Mathematics 22(2), 109–118 (1989)
6. 6.
Abu-Khzama, F.N.: A kernelization algorithm for d-hitting set. Journal of Computer and System Sciences 76(7), 524–531 (2010)
7. 7.
Gutin, G., Kim, E.J., Lampis, M., Mitsou, V.: Vertex cover problem parameterized above and below tight bounds. Theory of Computing Systems 48, 402–410 (2011)
8. 8.
Razgon, I., O’Sullivan, B.: Almost 2-SAT Is Fixed-Parameter Tractable (Extended Abstract). In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 551–562. Springer, Heidelberg (2008)
9. 9.
Flum, J., Grohe, M.: Parameterized complexity theory. Springer, Heidelberg (2006)
10. 10.
Cai, L., Juedes, D.: On the existence of subexponential parameterized algorithms. Journal of Computer and System Sciences 67, 789–807 (2003)
11. 11.
Berry, L.A., Kennedy, W.S., King, A.D., Li, Z., Reed, B.A.: Finding a maximum-weight induced k-partite subgraph of an i-triangulated graph. Discrete Applied Mathematics 158(7), 765–770 (2010)
12. 12.
Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: On multiway cut parameterized above lower bounds. To appear in Proceedings of IPEC 2011 (2011)Google Scholar
13. 13.
Grötschel, M., Lovász, L., Schrijver, A.: Geometric algorithms and combinatorial optimization. Springer, Heidelberg (1988)
14. 14.
Mahajan, M., Raman, V.: Parameterizing above guaranteed values: Maxsat and maxcut. Journal of Algorithms 31, 335–354 (1999)
15. 15.
Mahajan, M., Raman, V., Sikdar, S.: Parameterizing above or below guaranteed values. Journal of Computer and System Sciences 75(2), 137–153 (2009)
16. 16.
Wahlström, M.: Algorithms, measures and upper bounds for satisfiability and related problems. PhD thesis, Department of Computer and Information Science, Linköpings universitet, Sweden (2007)Google Scholar
17. 17.
Yannakakis, M., Gavril, F.: The maximum k-colorable subgraph problem for chordal graphs. Information Processing Letters 24(2), 133–137 (1987)
18. 18.
Golumbic, M.C.: Algorithmic graph theory and perfect graphs. Academic Press (1980)Google Scholar
19. 19.
Garey, M.R., Johnson, D.S.: Computers and intractability: A guide to the theory of NP-completeness. W.H.Freeman and Company (1979)Google Scholar
20. 20.
Impagliazzo, R., Paturi, R.: Complexity of k-sat. In: Proceedings of the 14th Annual IEEE Conference on Computational Complexity, pp. 237–240 (1999)Google Scholar
21. 21.
Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity. Journal of Computer and System Sciences 63(4), 512–530 (2001)
22. 22.
Niedermeier, R.: Invitation to fixed-parameter algorithms. Oxford University Press (2006)Google Scholar
23. 23.
Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer, Heidelberg (1999)
24. 24.
Mishra, S., Raman, V., Saurabh, S., Sikdar, S., Subramanian, C.R.: The Complexity of Finding Subgraphs Whose Matching Number Equals the Vertex Cover Number. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 268–279. Springer, Heidelberg (2007)
25. 25.
Mishra, S., Raman, V., Saurabh, S., Sikdar, S., Subramanian, C.R.: The complexity of könig subgraph problems and above-guarantee vertex cover. Algorithmica 61(4), 857–881 (2011)
26. 26.
Raman, V., Ramanujan, M.S., Saurabh, S.: Paths, Flowers and Vertex Cover. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 382–393. Springer, Heidelberg (2011)
27. 27.
Raman, V., Saurabh, S., Sikdar, S.: Efficient exact algorithms through enumerating maximal independent sets and other techniques. Theory of Computing Systems 41(30), 563–587 (2007)