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Between 2- and 3-colorability

Between 2- and 3-colorability . Rutgers University. The problem. O. Independent Set. G. X. Bipartite Graph. The problem. O. Independent Set. G. X. Bipartite Graph. tree. The problem. O. Independent Set. G. X. Bipartite Graph. tree. forest. The problem. O.

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Between 2- and 3-colorability

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  1. Between 2- and 3-colorability Rutgers University

  2. The problem O Independent Set G X Bipartite Graph

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

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

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

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

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

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

  9. Examples • Trees NP-complete • A. Brandstädt,V. B. Le, T. Szymczak, The complexity of some problems related to graph 3-colorability. Discrete Appl. Math.89 (1998) 59--73. • Forest NP-complete • Graphs of bounded vertex degree NP-complete • J. Kratochvíl, I.Schiermeyer, On the computational complexity of (O, P)-partition problems. Discuss. Math. Graph Theory17 (1997) 253--258.

  10. Examples • Trees NP-complete • A. Brandstädt,V. B. Le, T. Szymczak, The complexity of some problems related to graph 3-colorability. Discrete Appl. Math.89 (1998) 59--73. • Forest NP-complete • Graphs of bounded vertex degree NP-complete • J. Kratochvíl, I.Schiermeyer, On the computational complexity of (O, P)-partition problems. Discuss. Math. Graph Theory17 (1997) 253--258. • Complete bipartite Polynomial • A. Brandstädt, P.L. Hammer, V.B. Le, V. Lozin, Bisplit graphs. Discrete Math.299 (2005) 11--32.

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

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

  15. Speed of hereditary properties E.R. Scheinerman,J. Zito,On the size of hereditary classes of graphs. J. Combin. Theory Ser. B61 (1994) 16--39. Alekseev, V. E.On lower layers of a lattice of hereditary classes of graphs. (Russian) Diskretn. Anal. Issled. Oper. Ser. 14 (1997) 3--12. J. Balogh, B. Bllobás,D. Weinreich,The speed of hereditary properties of graphs. J. Combin. Theory Ser. B79 (2000) 131--156.

  16. Lower Layers • constant • polynomial • exponential • factorial

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

  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 • P1 The class of graphs of vertex degree at most 1

  20. Three minimal factorial classes of bipartite graphs • P1 The class of graphs of vertex degree at most 1 • P2 Bipartite complements to graphs in P1

  21. Three minimal factorial classes of bipartite graphs • P1 The class of graphs of vertex degree at most 1 • P2 Bipartite complements to graphs in P1 • P3 2K2-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)-partition 2-sat

  28. NP-complete results J. Kratochvíl, I.Schiermeyer, On the computational complexity of (O,P)-partition problems. Discuss. Math. Graph Theory17 (1997) 253--258. 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.Schiermeyer, On the computational complexity of (O,P)-partition problems. Discuss. Math. Graph Theory17 (1997) 253--258. 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. Node-deletion problems on bipartite graphs. SIAM J. Comput.10 (1981), no. 2, 310--327.

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

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