Graph Partitions

Partition V(G) into k sets (k=3)  Vertex partitions

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

Special properties of partitions Sets may be required to be independent

This kind of circle depicts an independent set

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

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

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

Deciding if a 2-colouring exists Algorithm succeeds 2-colouring existsNo odd cycles

Deciding if a 2-colouring exists Algorithm succeeds 2-colouring existsNo odd cycles

Deciding if a 2-colouring exists Algorithm succeeds 2-colouring existsNo odd cycles

G has a 2-colouring (is bipartite) if and only if it contains no induced

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

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

This is (corresponds to) a homomorphism. Here a homomorphism to C 5 - also known as a C 5 -colouring.

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 C 5 corresponds to a partition of V(G) into five independent sets with the right connections.

C5C5 f -1 (1) f -1 (2) f -1 (3) f -1 (4) f -1 (5)

Special properties of partitions Sets may be required to be cliques

This kind of circle depicts a clique

This is just a colouring of the complement of G

if it is partitionable as G is a split graph

 is in P Deciding if G is a split graph

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

Deciding if G is split Algorithm succeeds A splitting existsNo forbidden subgraphs [H-Klein-Nogueira-Protti]

(assuming all parts are nonempty) This is a clique cutset

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

G is a chordal graph if it contains no induced

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

G is a cograph if it contains no induced

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

This kind of line means all possible edges are present

Another well-known kind of partition A homogeneous set (module)

finding one is in P has applications in decomposition and recognition of comparability graphs (and in solving optimization problems on comparability graphs) [Gallai…]

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

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  =  [Chudnovsky, Robertson, Seymour, Thomas]

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]

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

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

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

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

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

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

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)

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

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

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]

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

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).

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

M has no 1’s (or no 0’s) [H-Nesetril, Feder-H-Huang] M has no *’s All PART(M) and LPART(M) in P [Feder-H] Classified PART(M)

CSP(H) Given a structure T with vertices V(H) and relations R 1 (H), … R k (H) of arities r 1, …, r k Decide whether or not an input structure G with vertices V(G) and relations R 1 (G), … R k (G), of the same arities r 1, …, r k admits a homomorphism f of G to H. DICHOTOMY CONJECTURE [Feder-Vardi] Each CSP(H) is in P or is NP-complete

If for every matrix M the problem PART(M) is in P or is NP-complete, then the Dichotomy Conjecture is true. [Feder-H] Thus hoping to classify all problems PART(M) appears to be overly ambitious… Can all PART(M) be classified?

G is complete bipartite if and only if it contains no induced

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

G is a bipartite graph if and only if It contains no induced

Another classification of PART(M)? For which matrices M can the problem PART(M) be described by finitely many forbidden induced subgraphs?

Infinitely many forbidden induced subgraphs occur whenever M contains or [Feder-H-Xie]

Do all others have finite sets of forbidden induced subgraphs? k l

NO…

For small matrices M If |M| ≤ 5, all other partition problems have only finitely many forbidden induced subgraphs If |M| = 6, there are other partition problems that have infinitely many forbidden induced subgraphs [Feder-H-Xie]

If |M| ≤ 5, all other partition problems have only finitely many forbidden induced subgraphs If |M| = 6, there are other partition problems that have infinitely many forbidden induced subgraphs [Feder-H-Xie] means without

Restrictions to inputs G Since these partitions relate closely to perfect graphs, we may want to restrict attention to (classes of) perfect graphs G

If M is normal The problem PART(M) restricted to perfect graphs G is in P [Feder-H] (fmfs)

BUT… …classifying PART(M), for perfect G, as being in P or being NP-complete, would still solve the dichotomy conjecture

If M is crossed The problem PART(M) restricted to chordal graphs G is in P [Feder-H-Klein-Nogueira-Protti]

BUT… …there are problems PART(M), restricted to chordal graphs G, which are NP-complete [Feder-H-Klein-Nogueira-Protti]

For all M A cograph G has a partition if and only if G does not contain one of a finite set of forbidden induced subgraphs [Feder-H-Hochstadter]

Are these problems CSP’s? Yes - two adjacent vertices of G have certain allowed images in H and two nonadjacent vertices of G have certain allowed images in H. (Two binary relations)

Are these problems CSP’s? Yes - two adjacent vertices of G have certain allowed images in H and two nonadjacent vertices of G have certain allowed images in H. (Two binary relations) No - this is not a CSP(T), as inputs are restricted to have each pair of distinct variables in a unique binary relation.

Full CSP’s Given a set L of positive integers, an L-full structure G has each k  L elements in a unique k-ary relation CSP L (H) is CSP(H) restricted to L-full structures G

Example with m binary relations Given a complete graph with edges coloured by 1, 2, …, m. Given such a G, colour the vertices 1, 2, …, m, without a monochromatic edge  i i i

 When m = 2, the problem is in P

 When m  4, it is NP-complete

 When m = 2, the problem is in P  When m  4, it is NP-complete  When m = 3, we only have algorithms of complexity n O ( log n / log log n ) [FHKS]

An algorithm of complexity n O(log n) solving the (more general) problem with lists Given a complete graph G with  edges coloured by 1, 2, 3, and  vertices equipped with lists  {1,2,3}

If all lists have size  2  Introduce a boolean variable for each vertex (use the first/second member of its list)  Express each edge-constraint as a clause of two variables Solve by 2-SAT

In general Let X be the set of vertices with lists {1,2,3} Recursively reduce |X| as follows:  Try to colour G without giving any vertex its majority colour  Give each vertex in turn its majority colour  (|X|-1) / 3 X X X

Analysis of Recursive Algorithm Time to solve problem with |X|= x T(x) = (1 + x T(2x/3)). T(2-SAT)

Analysis of Recursive Algorithm Time to solve problem with |X|= x T(x) = (1 + x T(2x/3)). T(2-SAT)  T(x) = x O(log x)

Analysis of Recursive Algorithm Time to solve problem with |X|= x T(x) = (1 + x T(2x/3)). T(2-SAT)  T(x) = x O(log x)  T(n) = n O(log n)

Can we say anything ? A kind of (quasi) dichotomy If 1  L then every CSP L (H) is  quasi-polynomial or  NP-complete [Feder-H]