Graph Partitions

1. Graph Partitions

2. Vertex partitions • Partition V(G) into k sets (k=3)

3. This kind of circle depicts an arbitrary set This kind of line means there may be edges between the two sets

4. Special properties of partitions • Sets may be required to be independent

5. This kind of circle depicts an independent set

6. This is just a k-colouring (k=3)

7. in P for k =1, 2  NP-complete for all other k Deciding if a k-colouring exists is (k=2)

8. 1 Obvious algorithm: Deciding if a 2-colouring exists 2 2 2 1 1 1 1 2 (k=2)

9. Deciding if a 2-colouring exists Algorithm succeeds No odd cycles 2-colouring exists

10. Deciding if a 2-colouring exists Algorithm succeeds No odd cycles 2-colouring exists

11. Deciding if a 2-colouring exists Algorithm succeeds No odd cycles 2-colouring exists

12. G has a 2-colouring(is bipartite) if and only if it contains no induced 7 . . . 3 5

13. Special properties of partitions  Sets may be required to have no edges joining them

14. This kind of dotted line means there are no edges joining the two sets

15. This is (corresponds to) a homomorphism. Here a homomorphism to C5 - also known as a C5-colouring.

16. A homomorphism of G to H(or an H-colouring of G) is a mapping f : V(G) V(H) such that uv E(G) implies f(u)f(v) E(H). A homomorphism f of G to C5 corresponds to a partition of V(G) into five independent sets with the right connections.

17. f -1(1) f -1(2) f -1(5) f -1(3) f -1(4) 1 2 5 3 C5 4

18. 1 2 5 3 4

19. Special properties of partitions • Sets may be required to be cliques

20. This kind of circle depicts a clique

21. This is just a colouring of the complement of G

22. G is a split graph if it is partitionable as

23. Deciding if G is a split graph  is in P

24. G is split graph if and only if it contains no induced 5 4

25. Deciding if G is split Algorithm succeeds No forbidden subgraphs A splitting exists [H-Klein-Nogueira-Protti]

26. This is a cliquecutset (assuming all parts are nonempty)

27. Deciding if G has a clique cutset • is in P • has applications in solving optimization problems on chordal graphs [Tarjan, Whitesides,…]

28. G is a chordal graph if it contains no induced 6 . . . 4 5

29. G is a chordal graph if it contains no induced 6 . . . 4 5 if and only if every induced subgraph is either a clique or has a clique cutset [Dirac]

30. G is a cograph if it contains no induced

31. G is a cograph if it contains no induced if and only if every induced subgraph is partitionable as or [Seinsche]

32. This kind of line means all possible edges are present

33. A homogeneousset (module) Another well-known kind of partition

34. A homogeneousset (module) • finding one is in P • has applications in decomposition and recognition of comparability graphs (and in solving optimization problems on comparability graphs) [Gallai…]

35. G is a perfect graph  =  holds for G and all its induced subgraphs.

36. G is a perfect graph  =  holds for G and all its induced subgraphs. G is perfect if and only if G and its complement contain no induced 7 . . . 3 5 [Chudnovsky, Robertson, Seymour, Thomas]

37. Perfect graphs • contain bipartite graphs, line graphs of bipartite graphs, split graphs, chordal graphs, cographs, comparability graphs • and their complements, and • model many max-min relations. [Berge]

38. Perfect graphs • contain bipartite graphs, line graphs of bipartite graphs, split graphs, chordal graphs, cographs, comparability graphs • and their complements, and • model many max-min relations. Basic graphs

39. G is perfect if and only if it is basic or it admits a partition … all others [Chudnovsky, Robertson, Seymour, Thomas]

40. Special properties of partitions Sets may be required to be • Independent sets • cliques • or unrestricted Between the sets we may require • no edges • all edges • or no restriction

41. The matrix M of a partition 0 if Vi is independent M(i,i) = 1 if Vi is a clique * if Vi is unrestricted 0 if Vi and Vj are not joined M(i,j) = 1 if Vi and Vj are fully joined * if Vi to Vj is unrestricted

42. The problem PART(M) • Instance: A graph G • Question: Does G admit a partition according to the matrix M ?

43. The problem SPART(M) • Instance: A graph G • Question: Does G admit a surjective partition according to M ? (the parts are non-empty)

44. The problem LPART(M) • Instance: A graph G, with lists • Question: Does G admit a list partition according to M ? (each vertex is placed to a set on its list)

45. For PART(M) we assume NO DIAGONAL ASTERISKS * M has a diagonal of k zeros and l ones ( k + l = n )

46. Small matrices M • When |M| ≤ 4: PART(M) classified as being in P or NP-complete [Feder-H-Klein-Motwani] • When |M| ≤ 4: SPART(M) classified as being in P or NP-complete [deFigueiredo-Klein-Gravier-Dantas] except for one matrix M

47. Small matrices M with lists • When |M| ≤ 4: LPART(M) classified as being in P or NP-complete, except for one matrix [Feder-H-Klein-Motwani] [de Figueiredo-Klein-Kohayakawa-Reed] [Cameron-Eschen-Hoang-Sritharan] • When |M| ≤ 3: digraph partition problems classified as being in P or NP-complete [Feder-H-Nally]

48. Classified PART(M) • M has no 1’s (or no 0’s) [H-Nesetril, Feder-H-Huang]

49. Classified PART(M) • M has no 1’s (or no 0’s) [H-Nesetril, Feder-H-Huang] PART(M) is in P if M corresponds to a graph which has a loop or is bipartite, and it is NP-complete otherwise LPART(M) is in P if M corresponds to a bi-arc graph, and it is NP-complete otherwise

50. Bi-Arc Graphs Defined as (complements of) certain intersection graphs… A common generalization of interval graphs (with loops) and (complements of) circular arc graphs of clique covering number two (no loops).