1. Between 2- and 3-colorability Rutgers University

2. The problem G X Bipartite Graph O Independent Set

3. The problem G X Bipartite Graph tree O Independent Set

4. The problem G X Bipartite Graph tree forest O Independent Set

5. The problem G X Bipartite Graph tree forest of bounded degree O Independent Set

6. The problem G X Bipartite Graph tree forest of bounded degree complete bipartite O Independent Set

7. Examples Trees NP-complete A. Brandstädt,A. Brandstädt, V. B. Le, T. Szymczak, The complexity of some problems related to graph 3-colorability. Discrete Appl. Math. 89 (1998) 59--73.V. B. Le, T. Szymczak,Discrete Appl. Math.

8. Examples Trees NP-complete A. Brandstädt,A. Brandstädt, V. B. Le, T. Szymczak, The complexity of some problems related to graph 3-colorability. Discrete Appl. Math. 89 (1998) 59--73.V. B. Le, T. Szymczak,Discrete Appl. Math. Forest NP-complete

9. Examples Trees NP-complete A. Brandstädt,A. Brandstädt, V. B. Le, T. Szymczak, The complexity of some problems related to graph 3-colorability. Discrete Appl. Math. 89 (1998) 59--73.V. B. Le, T. Szymczak,Discrete Appl. Math. Forest NP-complete Graphs of bounded vertex degree NP-complete J. Kratochvíl, I.J. Kratochvíl, I. Schiermeyer, On the computational complexity of (O, P)- partition problems. Discuss. Math. Graph Theory 17 (1997) 253--258.Schiermeyer,Discuss. Math. Graph Theory

10. Examples Trees NP-complete A. Brandstädt,A. Brandstädt, V. B. Le, T. Szymczak, The complexity of some problems related to graph 3-colorability. Discrete Appl. Math. 89 (1998) 59--73.V. B. Le, T. Szymczak,Discrete Appl. Math. Forest NP-complete Graphs of bounded vertex degree NP-complete J. Kratochvíl, I.J. Kratochvíl, I. Schiermeyer, On the computational complexity of (O, P)- partition problems. Discuss. Math. Graph Theory 17 (1997) 253--258.Schiermeyer,Discuss. Math. Graph Theory Complete bipartite Polynomial A. Brandstädt, P.L. Hammer, V.B. Le, V. Lozin,A. Brandstädt, P.L. Hammer, V.B. Le, V. Lozin, Bisplit graphs. Discrete Math. 299 (2005) 11--32.Discrete Math.

11. Question Is there any boundary separating difficult instances of the (O,P)-partition problem from polynomially solvable ones?

12. Question Is there any boundary separating difficult instances of the (O,P)-partition problem from polynomially solvable ones? Yes ?

13. Hereditary classes of graphs Definition. A class of graphs P is hereditary if X  P implies X-v  P for any vertex v  V(X)

14. Hereditary classes of graphs Definition. A class of graphs P is hereditary if X  P implies X-v  P for any vertex v  V(X) Examples: perfect graphs (bipartite, interval, permutation graphs), planar graphs, line graphs, graphs of bounded vertex degree.

15. Speed of hereditary properties E.R. Scheinerman,E.R. Scheinerman, J. Zito, On the size of hereditary classes of graphs. J. Combin. Theory Ser. B 61 (1994) 16--39.J. Zito,J. Combin. Theory Ser. B Alekseev, V. E.Alekseev, V. E. On lower layers of a lattice of hereditary classes of graphs. (Russian) Diskretn. Anal. Issled. Oper. Ser. 1 4 (1997) 3--12.Diskretn. Anal. Issled. Oper. Ser. 1 J. Balogh, B. Bllobás,J. Balogh, B. Bllobás, D. Weinreich, The speed of hereditary properties of graphs. J. Combin. Theory Ser. B 79 (2000) 131--156.D. Weinreich,J. Combin. Theory Ser. B

16. Lower Layers constant polynomial exponential factorial

17. Lower Layers constant polynomial exponential factorial permutation graphs line graphs graphs of bounded vertex degree graphs of bounded tree-width planar graphs

18. Minimal Factorial Classes of graphs  Bipartite graphs 3 subclasses  Complements of bipartite graphs 3 subclasses  Split graphs, i.e., graphs partitionable into an independent set and a clique 3 subclasses

19. Three minimal factorial classes of bipartite graphs  P 1 The class of graphs of vertex degree at most 1

20. Three minimal factorial classes of bipartite graphs  P 1 The class of graphs of vertex degree at most 1  P 2 Bipartite complements to graphs in P 1

21. Three minimal factorial classes of bipartite graphs  P 1 The class of graphs of vertex degree at most 1  P 2 Bipartite complements to graphs in P 1  P 3 2K 2 -free bipartite graphs (chain or difference graphs)

22. (O,P)-partition problem Let P be a hereditary class of bipartite graphs Problem. Determine whether a graph G admits a partition into an independent set and a graph in the class P

23. (O,P)-partition problem Let P be a hereditary class of bipartite graphs Problem. Determine whether a graph G admits a partition into an independent set and a graph in the class P Conjecture If P contains one of the three minimal factorial classes of bipartite graphs, then the (O,P)- partition problem is NP-complete. Otherwise it is solvable in polynomial time.

24. Polynomial-time results Theorem. If P contains none of the three minimal factorial classes of bipartite graphs, then the (O,P)- partition problem can be solved in polynomial time.

25. Polynomial-time results Theorem. If P contains none of the three minimal factorial classes of bipartite graphs, then the (O,P)- partition problem can be solved in polynomial time. If P contains none of the three minimal factorial classes of bipartite graphs, then P belongs to one of the lower layers exponential polynomial constant

26. Exponential classes of bipartite graphs Theorem. For each exponential class of bipartite graphs P, there is a constant k such that for any graph G in P there is a partition of V(G) into at most k independent sets such that every pair of sets induces either a complete bipartite or an empty (edgeless) graph.

27. Exponential classes of bipartite graphs Theorem. For each exponential class of bipartite graphs P, there is a constant k such that for any graph G in P there is a partition of V(G) into at most k independent sets such that every pair of sets induces either a complete bipartite or an empty (edgeless) graph. (O,P)-partition2-sat

28. NP-complete results J. Kratochvíl, I.J. Kratochvíl, I. Schiermeyer, On the computational complexity of (O,P)- partition problems. Discuss. Math. Graph Theory 17 (1997) 253--258.Schiermeyer,Discuss. Math. Graph Theory Theorem. If P is a monotone class of graphs different from the class of empty (edgeless) graphs, then the (O,P)-partition problem is NP-complete.

29. NP-complete results J. Kratochvíl, I.J. Kratochvíl, I. Schiermeyer, On the computational complexity of (O,P)- partition problems. Discuss. Math. Graph Theory 17 (1997) 253--258.Schiermeyer,Discuss. Math. Graph Theory Theorem. If P is a monotone class of graphs different from the class of empty (edgeless) graphs, then the (O,P)-partition problem is NP-complete. Corollary. The (O,P)-partition problem is NP-complete if P is the class of graphs of vertex degree at most 1.

30. One more result Yannakakis, M.Yannakakis, M. Node-deletion problems on bipartite graphs. SIAM J. Comput. 10 (1981), no. 2, 310--327. SIAM J. Comput.

31. One more result Yannakakis, M.Yannakakis, M. Node-deletion problems on bipartite graphs. SIAM J. Comput. 10 (1981), no. 2, 310--327. SIAM J. Comput. Let P be a hereditary class of bipartite graphs Problem*(P). Given a bipartite graph G, find a maximum induced subgraph of G belonging to P.

32. One more result Yannakakis, M.Yannakakis, M. Node-deletion problems on bipartite graphs. SIAM J. Comput. 10 (1981), no. 2, 310--327. SIAM J. Comput. Let P be a hereditary class of bipartite graphs Problem*(P). Given a bipartite graph G, find a maximum induced subgraph of G belonging to P. Theorem. If P is a non-trivial hereditary class of bipartite graphs containing one of the three minimal factorial classes of bipartite graphs, then Problem*(P) is NP-hard. Otherwise, it is solvable in polynomial time.

33. Thank you