# Simple Max-Cut for Split-Indifference Graphs and Graphs with Few P4’s

• Hans L. Bodlaender
• Celina M. H. de Figueiredo
• Marisa Gutierrez
• Ton Kloks
• Rolf Niedermeier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3059)

## Abstract

The simple max-cut problem is as follows: given a graph, find a partition of its vertex set into two disjoint sets, such that the number of edges having one endpoint in each set is as large as possible. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. The simple max-cut decision problem is known to be NP-complete for split graphs. An indifference graph is the intersection graph of a set of unit intervals of the real line. We show that the simple max-cut problem can be solved in linear time for a graph that is both split and indifference. Moreover, we also show that for each constant q, the simple max-cut problem can be solved in polynomial time for (q,q-4)-graphs. These are graphs for which no set of at most q vertices induces more than q-4 distinct P 4’s.

## Keywords

Maximal Clique Intersection Graph Interval Graph Decomposition Tree Graph Class
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Arbib, C.: A polynomial characterization of some graph partitioning problem. Inform. Process. Lett. 26, 223–230 (1987/1988)Google Scholar
2. 2.
Babel, L.: On the P4-structure of graphs, Habilitationsschrift, Zentrum für Mathematik, Technische Universität München (1997)Google Scholar
3. 3.
Babel, L., Kloks, T., Kratochvíl, J., Kratsch, D., Müller, H., Olariu, S.: Efficient algorithms for graphs with few P4’s. Combinatorics (Prague, 1998). Discrete Math. 235, 29–51 (2001)
4. 4.
Babel, L., Olariu, S.: On the structure of graphs with few P4s. Discrete Appl. Math. 84, 1–13 (1998)
5. 5.
Baumann, S.: A linear algorithm for the homogeneous decomposition of graphs, Report No. M-9615, Zentrum für Mathematik, Technische Universität München (1996)Google Scholar
6. 6.
Bodlaender, H.L., Jansen, K.: On the complexity of the maximum cut problem. In: Enjalbert, P., Mayr, E.W., Wagner, K.W. (eds.) STACS 1994. LNCS, vol. 775, pp. 769–780. Springer, Heidelberg (1994); Also in Nordic J. Comput. 7(1),14–31 (2000)Google Scholar
7. 7.
Bodlaender, H.L., Kloks, T., Niedermeier, R.: Simple max-cut for unit interval graphs and graphs with few P4’s. In: Extended abstracts of the 6th Twente Workshop on Graphs and Combinatorial Optimization, pp. 12–19 (1999); Also in Electronic Notes in Discrete Mathematics 3 (1999)Google Scholar
8. 8.
Bogart, K.P., West, D.B.: A short proof that ‘proper = unit’. Discrete Math. 201, 21–23 (1999)
9. 9.
Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: a Survey. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1999)Google Scholar
10. 10.
de Figueiredo, C.M.H., Meidanis, J., de Mello, C.P.: A linear-time algorithm for proper interval graph recognition. Inform. Process. Lett. 56, 179–184 (1995)
11. 11.
Eskin, E., Halperin, E., Karp, R.M.: Large scale reconstruction of haplotypes from genotype data. In: RECOMB 2003, pp. 104–113. ACM Press, New York (2003)Google Scholar
12. 12.
Giakoumakis, V., Roussel, F., Thuillier, H.: On P4-tidy graphs. Discrete Mathematics and Theoretical Computer Science 1, 17–41 (1997)
13. 13.
Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995)
14. 14.
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)
15. 15.
Gramm, J., Hirsch, E.A., Niedermeier, R., Rossmanith, P.: Worst-case upper bounds for MAX-2-SAT with application to MAX-CUT. Discrete Appl. Math. 130(2), 139–155 (2003)
16. 16.
Hadlock, F.O.: Finding a maximum cut of a planar graph in polynomial time. SIAM J. Comput. 4, 221–225 (1975)
17. 17.
Jamison, B., Olariu, S.: A tree representation for P4-sparse graphs. Discrete Appl. Math. 35, 115–129 (1992)
18. 18.
Jamison, B., Olariu, S.: Recognizing P4-sparse graphs in linear time. SIAM J. Comput. 21, 381–406 (1992)
19. 19.
Jamison, B., Olariu, S.: Linear time optimization algorithms for P4-sparse graphs. Discrete Appl. Math. 61, 155–175 (1995)
20. 20.
Jamison, B., Olariu, S.: p-components and the homogeneous decomposition of graphs. SIAM J. Discrete Math. 8, 448–463 (1995)
21. 21.
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thather, J.W. (eds.) Complexity of computation, pp. 85–103 (1972)Google Scholar
22. 22.
Kloks, T., Tan, R.B.: Bandwidth and topological bandwidth of graphs with few P4’s. In: 1st Japanese-Hungarian Symposium for Discrete Mathematics and its Applications, Kyoto, pp. 117–133 (1999); Discrete Appl. Math. 115(1-3), 117–133 (2001)Google Scholar
23. 23.
McKee, T.A., Morris, F.R.: Topics in Intersection Graph Theory. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1999)Google Scholar
24. 24.
Orlova, G.I., Dorfman, Y.G.: Finding the maximal cut in a graph. Engrg. Cybernetics 10, 502–504 (1972)
25. 25.
Ortiz, C., Maculan, Z.N., Szwarcfiter, J.L.: Characterizing and edge-colouring split-indifference graphs. Discrete Appl. Math. 82(1-3), 209–217 (1998)
26. 26.
Poljak, S., Tuza, Z.: Maximum cuts and large bipartite subgraphs. In: Cook, W., Lovász, L., Seymour, P. (eds.) DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Providence, RI. Amer. Math. Soc., vol. 20, pp. 181–244 (1995)Google Scholar
27. 27.
Rizzi, R., Bafna, V., Istrail, S., Lancia, G.: Practical algorithms and fixedparameter tractability for the single individual SNP haplotypying problem. In: Guigó, R., Gusfield, D. (eds.) WABI 2002. LNCS, vol. 2452, pp. 29–43. Springer, Heidelberg (2002)
28. 28.
Roberts, F.S.: On the compatibility between a graph and a simple order. J. Combinatorial Theory Ser. B 11, 28–38 (1971)
29. 29.
Wimer, T.V.: Linear algorithms on k-terminal graphs, PhD Thesis, Department of Computer Science, Clemson University, South Carolina (1987)Google Scholar

## Authors and Affiliations

• Hans L. Bodlaender
• 1
• Celina M. H. de Figueiredo
• 2
• Marisa Gutierrez
• 3
• Ton Kloks
• 4
• Rolf Niedermeier
• 5
1. 1.Institute of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands
2. 2.Instituto de Matemática and COPPEUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
3. 3.Departamento de MatemáticaUniversidad Nacional de La Plata, C. C. 172 (1900)La PlataArgentina
4. 4.
5. 5.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenGermany