Graph Partitions

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Graph Partitions
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• Vertex partitions

• Partition V(G) into k sets

(k3)
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This kind of circle depicts an arbitrary set
This kind of line means there may be edges
between the two sets
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Special properties of partitions

• Sets may be required to be independent

5
This kind of circle depicts an independent set
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This is just a k-colouring
(k3)
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Deciding if a k-colouring exists is

• in P for k 1, 2
• ? NP-complete for all other k

(k2)
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Deciding if a 2-colouring exists
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Obvious algorithm
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2
2
1
1
1
1
2
(k2)
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Deciding if a 2-colouring exists
Algorithm succeeds
2-colouring exists
No odd cycles
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Deciding if a 2-colouring exists
Algorithm succeeds
2-colouring exists
No odd cycles
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Deciding if a 2-colouring exists
Algorithm succeeds
2-colouring exists
No odd cycles
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G has a 2-colouring(is bipartite)

• if and only if it contains no induced

7 . . .
3
5
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Special properties of partitions

• ? Sets may be required to have no edges joining
  them

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This kind of dotted line means there are no
edges joining the two sets
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• This is (corresponds to) a homomorphism.
• Here a homomorphism to C5 - also known
• as a C5-colouring.

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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.

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f -1(1)
f -1(2)
f -1(3)
f -1(5)
f -1(4)
1
2
3
5
C5
4
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1
2
3
5
4
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Special properties of partitions

• Sets may be required to be cliques

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This kind of circle depicts a clique
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• This is just a colouring of the
• complement of G

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G is a split graph

• if it is partitionable as

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Deciding if G is a split graph

• ? is in P

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• G is split graph
• if and only if
• it contains no induced

5
4
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Deciding if G is split
Algorithm succeeds
A splitting exists
No forbidden subgraphs
H-Klein-Nogueira-Protti
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This is a clique cutset

• (assuming all parts are nonempty)

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Deciding if G has a clique cutset

• is in P
• has applications in solving optimization problems
  on chordal graphs
• Tarjan, Whitesides,

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G is a chordal graph

• if it contains no induced

6 . . .
4
5
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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
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G is a cograph

• if it contains no induced

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G is a cograph

• if it contains no induced

if and only if every induced subgraph is
partitionable as or
Seinsche
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This kind of line means all possible edges are
present
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A homogeneous set (module)

• Another well-known kind of partition

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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

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G is a perfect graph
? ?

• holds for G and all its induced subgraphs.

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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
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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

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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
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G is perfect

• if and only if it is basic or it admits a
• partition
• all others

Chudnovsky, Robertson, Seymour, Thomas
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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

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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

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The problem PART(M)

• Instance A graph G
• Question Does G admit a partition according to
  the matrix M ?

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The problem SPART(M)

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

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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)

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For PART(M) we assume

• NO DIAGONAL ASTERISKS
• M has a diagonal of k zeros and l ones
• ( k l n )

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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

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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

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Classified PART(M)

• M has no 1s (or no 0s)
• H-Nesetril, Feder-H-Huang

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Classified PART(M)

• M has no 1s (or no 0s)
• 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

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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).

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Classified PART(M)

• M has no 1s (or no 0s)
• H-Nesetril, Feder-H-Huang
• M has no s

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Classified PART(M)

• M has no 1s (or no 0s)
• H-Nesetril, Feder-H-Huang
• M has no s
• All PART(M) and LPART(M) in P Feder-H

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CSP(H)

• Given a structure T with vertices V(H) and
  relations R1(H), Rk(H) of arities r1, , rk
• Decide whether or not an input structure G with
  vertices V(G) and relations R1(G), Rk(G), of
  the same arities r1, , rk admits a homomorphism
  f of G to H.
• DICHOTOMY CONJECTURE Feder-Vardi
• Each CSP(H) is in P or is NP-complete

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Can all PART(M) be classified?

• 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

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G is complete bipartite

• if and only if it contains no induced

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G is a split graph

• if and only if
• it contains no induced

5
4
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G is a bipartite graph

• if and only if
• It contains no induced

7 . . .
3
5
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Another classification of PART(M)?

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

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Infinitely many forbidden induced subgraphs occur

• whenever M contains
• or
• Feder-H-Xie

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

• NO

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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

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means without

• 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

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Restrictions to inputs G

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

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If M is normal

• The problem PART(M) restricted to
• perfect graphs G is in P
• Feder-H
• (fmfs)

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BUT

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

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If M is crossed

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

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BUT

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

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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

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Are these problems CSPs?

• 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)

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Are these problems CSPs?

• 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.

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Full CSPs

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

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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
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• When m 2, the problem is in P

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• When m 2, the problem is in P
• When m ? 4, it is NP-complete

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• 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

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• An algorithm of complexity nO(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

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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

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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
? (X-1) / 3
X
X
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Analysis of Recursive Algorithm

• Time to solve problem with X x
• T(x) (1 x T(2x/3)) . T(2-SAT)

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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)

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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)

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Can we say anything ?

• A kind of (quasi) dichotomy
• If 1 ? L then every CSPL(H) is
• ? quasi-polynomial or
• ? NP-complete
• Feder-H